Enumerating submonoids of finite commutative monoids

Caoilainn Kirkpatrick; Amelie el Mahmoud; Kyle Ormsby; Ang\'elica M. Osorno; Dale Schandelmeier-Lynch; Riley Shahar; Lixing Yi; Avery Young; Saron Zhu

arXiv:2508.20786·math.CO·August 29, 2025

Enumerating submonoids of finite commutative monoids

Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Ang\'elica M. Osorno, Dale Schandelmeier-Lynch, Riley Shahar, Lixing Yi, Avery Young, Saron Zhu

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TL;DR

This paper develops a method to enumerate submonoids of finite commutative monoids extended by a max-structured set, revealing a finite sum formula for their counts when the monoid is idempotent, with applications in algebra and topology.

Contribution

It introduces a transfer matrix approach for enumeration and establishes a finite sum formula for submonoid counts in idempotent cases, answering Knuth's question.

Findings

01

Enumeration via transfer matrix method.

02

Finite sum formula for submonoid counts in idempotent monoids.

03

Applications to saturated transfer systems in topology.

Abstract

Given a finite commutative monoid $M$ , we show that submonoids of $M \times [n]$ - where $[n] = {0, 1, \dots, n}$ is equipped with the max operation $\lor$ - may be enumerated via the transfer matrix method. When $M$ is also idempotent, we show that there are finitely many integers $λ$ and rational numbers $b_{λ}$ (only depending on $M$ ) such that the number of submonoids of $M \times [n]$ is $\sum_{λ} b_{λ} λ^{n}$ . This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.

Tables1

Table 1. Table 1 . Several finite join-semilattices P P , and the associated eigenvalues and normalized coefficients b ^ λ := b λ ∏ μ ≠ λ ( λ − μ ) \hat{b}_{\lambda}:=b_{\lambda}\prod_{\mu\neq\lambda}(\lambda-\mu) , as discussed in Section 3.1 . The cases included are either products of chains, lattices M k = { ⊥ , 1 , 2 , … , k , ⊤ } M_{k}=\{\bot,1,2,\ldots,k,\top\} with integer elements incomparable, or the pentagon lattice N 5 N_{5} .

P	$λ$	$b_{λ}$	${\hat{b}}_{λ}$
$[1] \times [1]$	2	1/2	4
	3	1	-3
	4	-12	-48
	6	35/2	-420
$[2] \times [1]$	2	-13/48	-1560
	3	-111/35	2664
	4	14	4032
	5	16	-2880
	6	315/8	7560
	7	-154	55440
	8	-406/15	-38976
	10	15471/112	-5569560
$[3] \times [1]$	2	7213/34320	109060560
	3	28631/8400	-148423104
	4	-101729/13860	-58595904
	5	-797/15	128540160
	6	719/24	32613840
	7	-45157/120	260104320
	8	59599/75	480606336
	9	1482/5	-215115264
	10	735867/560	1589472720
	11	-840037/336	7257919680
	12	-42493/60	-7709929920
	13	-525746/2475	16958462976
	15	12800093/8580	9289795495680
$[4] \times [1]$	2	-335766173/1551950400	-1462113946298880
	3	-5004577/1501500	1255261295296512
	4	32846000389/3216213000	454063109377536
	5	9458167/96096	-823723113830400
	6	-423949649/960960	-988989741187200
	7	799421317/831600	-773584020034560
	8	250240977/143000	653813609283072
	9	-17901851/5250	748364969189376
	10	22150545/4928	723419079264000
	11	-796666013/23520	4955899933670400
	12	2096204/75	4600535229480960
	13	45969602783/4158000	-2541935155488768
	14	15454724597/1848000	3364926277055616
	15	21650713/572	-33940558368460800
	16	-26792689523/540540	-129633748988083200
	17	-1273644454/75075	177477158195527680
	18	-147245784937/21021000	-439673557889323008
	19	-168120903799/68068000	1757022150832736256
	21	665038415449/31039008	2606354714139837696000
${[1]}^{3}$	2	2453/12480	2649240
	3	-8621/21420	620712
	4	-351/32	-4447872
	5	108/175	-116640
	6	-13385/288	-6746040
	7	96831/520	-34859160
	8	493/10	23853312
	10	-2956707/11200	1064414520
	12	-31273/2520	-360264960
	15	-3336709/23400	92493573480
	20	159923969/530400	19344403290240
$M_{3}$	2	3/16	12
	3	3/7	-9
	4	0	0
	6	-105/8	1260
	10	2745/112	32940
$M_{4}$	2	1/64	16
	3	3/35	-27
	4	12/7	576
	6	-35/8	5040
	10	-2745/224	-131760
	18	80223/2240	-11552112
$M_{5}$	2	-125/2048	-2000
	3	-19/217	855
	4	12/7	17280
	6	-25/32	25200
	10	-13725/1792	-1976400
	18	-80223/7168	57760560
	34	3559545/63488	17940106800
$M_{6}$	2	-5241/65536	-167712
	3	-225/1519	91125
	4	36/31	725760
	6	75/64	-2268000
	10	-233325/50176	-67197600
	18	-80223/8192	2425943520
	34	-10678635/1015808	-107640640800
	66	9342726965/99549184	-121081741466400
$M_{7}$	2	-574609/8388608	-18387488
	3	-3933/27559	11150055
	4	110/217	39916800
	6	16205/7936	-490039200
	10	-112545/57344	-3403360800
	18	-7781631/917504	235316521440
	34	-373752225/32505856	-11302267284000
	66	-9342726965/910163968	847572190264800
	130	38345903755445/231181647872	3478740388693970400
$N_{5}$	2	-1/3	-48
	3	11/5	-66
	4	12	192
	5	-20	360
	6	-35	-1680
	8	812/15	-38976

Equations147

# SubMon (M \times [n]) = A, B \in SubMon M \sum (W^{n})_{A, B} .

# SubMon (M \times [n]) = A, B \in SubMon M \sum (W^{n})_{A, B} .

W_{A, B} = # {I \subseteq A ∣ I an ideal of A and I \cup B = A} .

W_{A, B} = # {I \subseteq A ∣ I an ideal of A and I \cup B = A} .

# SubMon (M \times [n]) = 2^{n} \cdot m \geq 0 \sum c_{m} 2^{- m} (m n) \sim \frac{1}{h !} c_{h} n^{h} 2^{n - h},

# SubMon (M \times [n]) = 2^{n} \cdot m \geq 0 \sum c_{m} 2^{- m} (m n) \sim \frac{1}{h !} c_{h} n^{h} 2^{n - h},

# SubMon (M \times [n]) = λ \in Λ \sum b_{λ} λ^{n} .

# SubMon (M \times [n]) = λ \in Λ \sum b_{λ} λ^{n} .

# SubSemi \prod [n_{i}] = 2# SubMon \prod [n_{i}] .

# SubSemi \prod [n_{i}] = 2# SubMon \prod [n_{i}] .

k = \frac{1}{2} B_{9, 9} = 22111390122811;

k = \frac{1}{2} B_{9, 9} = 22111390122811;

Sub C_{N} ≅ [n_{1}] \times \dots \times [n_{r}] .

Sub C_{N} ≅ [n_{1}] \times \dots \times [n_{r}] .

I_{n} \subseteq I_{n - 1} \subseteq \dots \subseteq I_{1} \subseteq I_{0} = N

I_{n} \subseteq I_{n - 1} \subseteq \dots \subseteq I_{1} \subseteq I_{0} = N

I_{k} I_{ℓ} \subseteq I_{k \lor ℓ} ∖ I_{(k \lor ℓ) + 1}

I_{k} I_{ℓ} \subseteq I_{k \lor ℓ} ∖ I_{(k \lor ℓ) + 1}

w (A, B) := # {I \subseteq A ∣ I an ideal of A and I \cup B = A} .

w (A, B) := # {I \subseteq A ∣ I an ideal of A and I \cup B = A} .

W_{A, B} = w (A, B) .

W_{A, B} = w (A, B) .

m_{A} (n) := # {R \leq M \times [n] ∣ π_{n} R = A} .

m_{A} (n) := # {R \leq M \times [n] ∣ π_{n} R = A} .

m_{A} (0) = 1 and m_{A} (n + 1) = B \leq M \sum w (A, B) m_{B} (n) .

m_{A} (0) = 1 and m_{A} (n + 1) = B \leq M \sum w (A, B) m_{B} (n) .

# SubMon (M \times [n]) = A, B \leq P \sum (W^{n})_{A, B} .

# SubMon (M \times [n]) = A, B \leq P \sum (W^{n})_{A, B} .

# SubMon (M \times [n]) = Θ (n^{k} i (M)^{n}) .

# SubMon (M \times [n]) = Θ (n^{k} i (M)^{n}) .

# SubMon (M \times [n]) \sim C \cdot i (M)^{n}

# SubMon (M \times [n]) \sim C \cdot i (M)^{n}

# SubMon (G \times [n]) = 2^{n} \cdot m \geq 0 \sum c_{m} 2^{- m} (m n) \sim \frac{1}{h !} c_{h} n^{h} 2^{n - h},

# SubMon (G \times [n]) = 2^{n} \cdot m \geq 0 \sum c_{m} 2^{- m} (m n) \sim \frac{1}{h !} c_{h} n^{h} 2^{n - h},

w (A, B) = ⎩ ⎨ ⎧ 210 if A = B, if B < A, otherwise .

w (A, B) = ⎩ ⎨ ⎧ 210 if A = B, if B < A, otherwise .

W = 2 I_{k} + L

W = 2 I_{k} + L

W^{n} = m \geq 0 \sum (m n) 2^{n - m} L^{m} = 2^{n} \cdot m \geq 0 \sum 2^{- m} (m n) L^{m} .

W^{n} = m \geq 0 \sum (m n) 2^{n - m} L^{m} = 2^{n} \cdot m \geq 0 \sum 2^{- m} (m n) L^{m} .

# SubMon (G \times [n]) = 1^{⊤} W^{n} 1 = 2^{n} \cdot m \geq 0 \sum 2^{- m} (m n) 1^{⊤} L^{m} 1 .

# SubMon (G \times [n]) = 1^{⊤} W^{n} 1 = 2^{n} \cdot m \geq 0 \sum 2^{- m} (m n) 1^{⊤} L^{m} 1 .

# SubMon (G \times [n]) = 2^{n} \cdot m \geq 0 \sum c_{m} 2^{- m} (m n)

# SubMon (G \times [n]) = 2^{n} \cdot m \geq 0 \sum c_{m} 2^{- m} (m n)

F_{ij} (x) := n \geq 0 \sum (W^{n})_{ij} x^{n} .

F_{ij} (x) := n \geq 0 \sum (W^{n})_{ij} x^{n} .

F_{ij} (x) = \frac{( - 1 ) ^{i + j} det ( I - x W : j , i )}{det ( I - x W )}

F_{ij} (x) = \frac{( - 1 ) ^{i + j} det ( I - x W : j , i )}{det ( I - x W )}

F (x) = i, j \sum F_{i, j} (x)

F (x) = i, j \sum F_{i, j} (x)

F (x) = n \geq 0 \sum # SubMon (M \times [n]) x^{n} .

F (x) = n \geq 0 \sum # SubMon (M \times [n]) x^{n} .

F (x) = \frac{\sum _{i, j} ( - 1 ) ^{i + j} det ( I - x W : j , i )}{\prod _{A \leq M} ( 1 - w ( A , A ) x )},

F (x) = \frac{\sum _{i, j} ( - 1 ) ^{i + j} det ( I - x W : j , i )}{\prod _{A \leq M} ( 1 - w ( A , A ) x )},

SubMon P

SubMon P

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TopicsRings, Modules, and Algebras · Advanced Algebra and Logic · semigroups and automata theory

Full text

Enumerating submonoids of finite commutative monoids

Caoilainn Kirkpatrick

Reed College

[email protected]

,

Amelie el Mahmoud

Reed College

[email protected]

,

Kyle Ormsby

Reed College

[email protected]

,

Angélica M. Osorno

Reed College

[email protected]

,

Dale Schandelmeier-Lynch

Reed College

[email protected]

,

Riley Shahar

University of Pennsylvania

[email protected]

,

Lixing Yi

Reed College

[email protected]

,

Avery Young

Reed College

[email protected]

and

Saron Zhu

University of Michigan

[email protected]

Abstract.

Given a finite commutative monoid $M$ , we show that submonoids of $M\times[n]$ — where $[n]=\{0,1,\ldots,n\}$ is equipped with the max operation $\vee$ — may be enumerated via the transfer matrix method. When $M$ is also idempotent, we show that there are finitely many integers $\lambda$ and rational numbers $b_{\lambda}$ (only depending on $M$ ) such that the number of submonoids of $M\times[n]$ is $\sum_{\lambda}b_{\lambda}\lambda^{n}$ . This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.

1. Introduction

Subgroup lattices have long played an important role in the theory of finite groups. While $\operatorname{Sub}G$ is far from a complete invariant of a finite group $G$ , many important properties of $G$ are faithfully represented by the isomorphism type of $\operatorname{Sub}G$ . In light of this, subgroup lattices have enjoyed intense study; see [Sch94] for an overview.

In this paper, we turn our attention to a more general but less ubiquitous algebraic structure: monoids and their submonoid lattices, with particular emphasis on finite commutative and finite commutative idempotent monoids. See Section 2 for recollections of essential definitions.

The main problem we attack is easy to state. For a natural number $n$ , let $[n]=\{0,1,\ldots,n\}$ with the operation $k\vee\ell=\max\{k,\ell\}$ .

Question.

Given a finite monoid $M$ , to what extent does $\operatorname{SubMon}M$ determine the cardinality of $\operatorname{SubMon}(M\times[n])$ ?

Varying the hypotheses on $M$ , we arrive at several satisfactory answers. Our most general result employs the transfer matrix method, which is nicely summarized in [Sta11, Chapter 4]. The idea is that, when counted with appropriate weights, length $n$ walks in a particular weighted directed graph $G(M)$ count the submonoids of $M\times[n]$ . These weighted walks are computed by powers of the adjacency matrix $W=W(M)$ of $G(M)$ , thus allowing us to bring linear algebraic tools to bear on the problem.

Theorem (Section 2.1).

Suppose $M$ is a finite commutative monoid. Then there is a lower triangular matrix $W$ of natural numbers with rows and columns indexed by a linear extension of $\operatorname{SubMon}M$ such that

[TABLE]

The entries of $W$ are combinatorial in nature, with

[TABLE]

(See Section 2.1 and Section 2.1.) Carefully analyzing these numbers for $M$ a finite Abelian group results in a surprising corollary:

Theorem (Section 2.1).

Suppose $M$ is a finite Abelian group and $n$ is a natural number. Then

[TABLE]

where $c_{m}$ is the number of length $m$ chains in the subgroup lattice of $M$ and $h$ is the height of $\operatorname{Sub}M$ , i.e., the length of the longest chain in the subgroup lattice.

Specializing in another direction, we consider the case of finite idempotent monoids. These are equivalent to finite join-semilattices; see Section 2(b).

Theorem (Section 3.1, Section 3.1).

Suppose $M$ is a finite commutative idempotent monoid. Then the matrix $W(M)$ is diagonalizable. It follows that there is a finite set of positive integers $\Lambda=\Lambda(M)$ and rational numbers $b_{\lambda}=b_{\lambda}(M)$ for $\lambda\in\Lambda$ such that

[TABLE]

The above theorem provides a partial answer to a question of Don Knuth [Knu24, Problem 10]. Knuth asks how many “max-closed relations” there are on a finite product of chains $\prod[n_{i}]$ . His max-closed relations are just subsemigroups of $\prod[n_{i}]$ under join. It is straightforward to check that for a join-semilattice $P$ , the “remove $e$ ” map is two-to-one from subsemigroups to submonoids, and hence

[TABLE]

Knuth [Knu24] details how $\operatorname{SubSemi}([m]\times[n])$ is enumerated by the poly-Bernoulli numbers defined by Kaneko [Kan97]; see Section 3.3. Section 3.1 provides an inductive method for attacking this problem, but remains too unwieldy for Knuth’s stated goal of determining $\#\operatorname{SubSemi}([8]^{3})$ . Indeed, we would first need to diagonalize the matrix $W([8]^{2})$ which is $k\times k$ with

[TABLE]

given that storing such a matrix would require around 2 million zettabytes, this is simply too large for practical computation.

The connection between commutative idempotent monoids and join-semilattices provides additional motivation for our work. Indeed, we undertook this project not with monoids in mind, but rather with the goal of understanding saturated transfer systems on lattices. These are combinatorial structures on subgroup lattices that are important in equivariant infinite loop space theory; see Section 4 and [HMOO22, Rub21]. In this context, $M=\operatorname{Sub}G$ (for $G$ a finite Abelian group) equipped with the join operation. It is a consequence of [BCH*+*25] that saturated transfer systems on $\operatorname{Sub}G$ are in bijective correspondence with $\operatorname{SubMon}(\operatorname{Sub}G)$ .

For $G=C_{N}$ cyclic of order $N=p_{1}^{n_{1}}\cdots p_{r}^{n_{r}}$ with $p_{i}$ distinct primes and $n_{i}$ natural numbers,

[TABLE]

As such, the problem of enumerating saturated transfer systems on $\operatorname{Sub}C_{N}$ is equivalent to Knuth’s problem on max-closed relations on $\prod[n_{i}]$ , and Section 3.1 again provides traction.

*Remark 1.1**.*

Let $\mathrm{CMon}$ denote the category of commutative monoids and monoid homomorphisms (see Section 2 for definitions), and write $\mathrm{CMon}/[n]$ for the slice category of commutative monoids equipped with a homomorphism to $[n]$ . We can view objects $p\colon N\to[n]$ of $\mathrm{CMon}/[n]$ as strongly multiplicative length $n+1$ filtrations by ideals. Here an ideal $I$ is a subset $I\subseteq N$ satisfying $IN=I$ , and the filtration by ideals takes the form

[TABLE]

where $I_{k}=p^{-1}[k,n]$ . In order to reconstruct a monoid homomorphism $p\colon N\to[n]$ from such a chain of ideals, it must also satisfy the strongly multiplicative criterion

[TABLE]

where $k\vee\ell=\max\{k,\ell\}$ .

Observe that $M\times[n]$ is naturally an object of $\mathrm{CMon}/[n]$ via projection. In fact, the assignment $M\mapsto M\times[n]$ is left adjoint to the forgetful functor $\mathrm{CMon}/[n]\to\mathrm{CMon}$ , and thus we may view $M\times[n]$ as the free commutative monoid equipped with a strongly multiplicative length $n+1$ filtration by ideals. This provides a categorical justification for studying $M\times[n]$ and its submonoids beyond our applications to max-closed relations and saturated transfer systems.

Overview

We now briefly outline the rest of the paper. In Section 2, we review some basics on monoids and submonoids, introduce the (weighted, directed) submonoid graph of a finite commutative monoid $M$ , and use this to prove Section 2.1, our most general theorem enumerating submonoids of $M\times[n]$ . We specialize this theorem to $M$ a commutative Abelian group in Section 2.1, and conclude the section with some observations on generating functions.

In Section 3, we suppose that $M$ is finite, commutative, and idempotent, and use these assumptions to derive a particularly special form for $\#\operatorname{SubMon}(M\times[n])$ in this case; see Section 3.1. We use this to demonstrate that $\#\operatorname{SubMon}(M\times[n])$ is a constant-recursive sequence in this setting, and then make the constants in Section 3.1 explicit for a selection of finite commutative idempotent monoids. We conclude Section 3 by making the connection between our work, [Knu24], and poly-Bernoulli numbers explicit, including a conjecture regarding multivariable exponential generating functions.

In Section 4, we demonstrate how our techniques apply to the enumeration of saturated transfer systems on finite lattices. This makes our enumerations applicable in the theory of $N_{\infty}$ -operads and equivariant ring spectra [BH15, Rub21].

Acknowledgments

This work was completed by the summer 2024 edition of Reed College’s Collaborative Mathematics Research Group, supported by NSF award DMS–2204365.

2. Transition matrices for submonoids

Throughout this section, let $M=(M,\cdot)$ be a finite commutative monoid with identity element $e$ ; in other words, $M$ is a finite set equipped with binary operation $\cdot$ satisfying $x\cdot y=y\cdot x$ , $x\cdot(y\cdot z)=(x\cdot y)\cdot z$ , and $x\cdot e=x=e\cdot x$ for all $x,y,z\in M$ . A homomorphism between monoids $M,N$ is a function $f\colon M\to N$ satisfying $f(x\cdot y)=f(x)\cdot f(y)$ and $f(e)=e$ . We will often implicitly work in the category of monoids and monoid homomorphisms, adding adjectives like finite, commutative, or idempotent as necessary.

Example 2.1.

We briefly recall some important examples and fix notation:

(a)

Any commutative group is a commutative monoid. 2. (b)

Suppose $(P,\leq)$ is a finite join-semilattice, that is, a partially ordered set admitting least upper bounds (joins) $\vee$ . Then $(P,\vee)$ is a commutative idempotent ( $x\vee x=x$ ) monoid. These structures are in fact equivalent: if $(M,\cdot)$ is a commutative idempotent monoid, then $x\leq y\iff x\cdot y=y$ makes $(M,\leq)$ a join-semilattice. 3. (c)

As an important subexample of (b), for a natural number $n$ , the finite chain $[n]=\{0<1<\cdots<n\}$ is a commutative (idempotent) monoid under $\ell\vee m=\max\{\ell,m\}$ . 4. (d)

Given two commutative monoids $M,N$ , their Cartesian product $M\times N$ is a commutative monoid under $(x,y)\cdot(x^{\prime},y^{\prime})=(x\cdot x^{\prime},y\cdot y^{\prime})$ . 5. (e)

For a set $S$ , the power set of $S$ is a commutative (idempotent) monoid with operation given by set union. It is isomorphic to $[1]^{\#S}$ .

Definition 2.2.

Let $(M,\cdot)$ be a monoid. A submonoid $A$ of $M$ is a subset of $M$ that contains $e$ and is closed under the operation $\cdot$ . We denote the collection of submonoids of $M$ by $\operatorname{SubMon}M$ and write $A\leq M$ when $A$ is a submonoid of $M$ .

2.1. The submonoid graph

We now construct a weighted directed graph $G(M)$ associated with a commutative monoid $M$ . Its vertices will consist of submonoids of $M$ , and we will ultimately show that the total weight of length $n$ walks in $G(M)$ counts the number of submonoids of $M\times[n]$ .

Definition 2.3.

An ideal of a commutative monoid $M$ is a subset $I\subseteq M$ with the property that $I\cdot M=I$ .

*Remark 2.4**.*

If $M$ is also idempotent, then ideals of $M$ are the same as upwards closed subsets (upsets) of $(M,\leq)$ .

Definition 2.5.

Suppose $M$ is a finite commutative monoid. The submonoid graph of $M$ is the weighted directed graph $G(M)$ with vertices the submonoids of $M$ and weight function $w\colon\operatorname{SubMon}(M)\times\operatorname{SubMon}(M)\to\mathbb{N}$ given by

[TABLE]

After (sometimes implicitly) choosing a linear ordering of $\operatorname{SubMon}M$ , we write $W=W(M)$ for the associated adjacency matrix in which

[TABLE]

Example 2.6.

Let $M=[1]\times[1]$ . Let $A=\{(0,0),(0,1),(1,1)\}$ and $B=\{(0,0),(1,1)\}$ . We draw $B$ in black and $A\smallsetminus B$ in red:

[TABLE]

In order to count towards $w(M,N)$ , an ideal $I$ must contain all of $A\smallsetminus B=\{(0,1)\}$ , and because of the ideal condition, it must contain $(1,1)$ as well, so our only choice is to either include $(0,0)$ or not. Thus $w(M,N)=2$ . The entire submonoid graph of $M$ and its adjacency matrix are displayed in Figure 1.

We will show that length $n$ walks in $G(M)$ enumerate submonoids of $M\times[n]$ . Perhaps surprisingly, they do so in terms of the projection of a submonoid onto $M\times[0]\cong M$ . Given $n\in\mathbb{N}$ , let $\pi=\pi_{n}\colon M\times[n]\to M$ denote the projection map $\pi(x,k)=x$ . The map $\pi$ is a homomorphism of monoids and thus takes submonoids to submonoids.

Lemma 2.7.

Let $M$ be a finite commutative monoid. For $A$ a submonoid of $M$ , let

[TABLE]

Then

[TABLE]

Proof.

The proof proceeds in the following steps:

Check the base case for $M\times[0]\cong M$ . 2. 2.

From each submonoid $R\leq M\times[n+1]$ such that $\pi_{n+1}R=A$ , construct a submonoid $R^{\prime}$ of $M\times[n]$ together with an ideal $I$ of $A$ such that $I\cup\pi_{n}R^{\prime}=A$ . 3. 3.

Construct an inverse assignment taking the data of $R^{\prime}\leq M\times[n]$ and an ideal $I$ of $A$ such that $I\cup\pi_{n}R^{\prime}=A$ to a submonoid of $M\times[n+1]$ that projects to $A$ . 4. 4.

Observe that by partitioning on $B=\pi(R^{\prime})$ , the bijection from the previous two steps gives the inductive formula.

Step 1 is immediate; Step 4 follows from Steps 2 and 3.

For Step 2, let $R\leq M\times[n+1]$ be a submonoid such that $\pi_{n+1}R=A$ . Define $R^{\prime}=R\cap(M\times[n])$ and $I=\pi_{n+1}(R\cap(M\times\{n+1\}))$ . It is straightforward to check that $I$ is an ideal of $A$ such that $I\cup\pi_{n}R^{\prime}=A$ .

For Step 3, let $R^{\prime}$ be a submonoid of $M\times[n]$ and $I\subseteq A$ be an ideal of $A$ with $I\cup\pi_{n}R^{\prime}=A$ . We set $R:=R^{\prime}\cup(I\times\{n+1\})$ . It is straightforward to check that $R$ is a submonoid of $M\times[n+1]$ with $\pi_{n+1}R=A$ .

Additionally, the assignments of Steps 2 and 3 are inverse by construction, concluding the proof. ∎

The previous lemma establishes the following theorem.

Theorem 2.8.

Let $M$ be a finite commutative monoid, suppose $A,B\in\operatorname{SubMon}(M)$ , and let $W=W(M)$ be the adjacency matrix of the submonoid graph $G(M)$ with respect to some linear ordering of $\operatorname{SubMon}(M)$ . Then the number of submonoids $R$ of $M\times[n]$ with $\pi_{n}R=A$ and $\pi_{n-1}(R\cap(M\times[n-1]))=B$ is the $(A,B)$ -entry of $W^{n}$ , and the total number of submonoids of $M\times[n]$ is

[TABLE]

∎

In the language of [Sta11, Chapter 4], the above theorem establishes that $W$ is the transfer matrix for submonoids of $M\times[n]$ . We can use it to establish asymptotics for $\#\operatorname{SubMon}(M\times[n])$ in terms of the number of ideals of submonoids of $M$ . To do so, we will need to make some observations regarding the shape of $W(M)$ .

To fix terminology, recall that for a partially ordered set $P$ , a linear extension of $P$ is a total ordering of $P$ that respects the partial order on $P$ .

Lemma 2.9.

Suppose $M$ is a finite commutative monoid and fix a linear extension of $\operatorname{SubMon}(M)$ under inclusion. When written with respect to this ordering, $W(M)$ is lower triangular.

Proof.

This is simply the observation that $B\not\subseteq A$ implies that $w(A,B)=0$ . ∎

To state the following theorem on the asymptotic growth of $\#\operatorname{SubMon}(M\times[n])$ , we establish the following notation: let $i(M)$ denote the maximal number of ideals of a submonoid of $M$ ; let $r(M)$ denote the number of submonoids $A\leq M$ for which $w(A,A)=i(M)$ . (We warn the reader that at this level of generality, $i(M)$ may not be equal to $w(M,M)$ , but it is equal to the maximal value of $w(A,A)$ for $A\leq M$ . When $M$ is idempotent, Section 3.1 implies that $i(M)=w(M,M)$ and $r(M)=1$ .)

Theorem 2.10.

If $M$ is a finite commutative monoid, then asymptotically in $n$ ,

[TABLE]

for some $0\leq k\leq r(M)-1$ . If $r(M)=1$ , then

[TABLE]

for $C=C(M)$ a positive constant.

Proof.

We leave the details to the reader, but note that this follows from a standard analysis of the Jordan decomposition of the lower triangular matrix $W(M)$ (Section 2.1) with nonnegative entries; see for instance [LeV07, Appendix D.2]. The crucial observations are that $i(M)$ is the largest eigenvalue of $W(M)$ and $k$ is the length of the longest path in the induced subgraph (not including self-loops) on $A\leq M$ such that $w(A,A)=i(M)$ . ∎

We conclude this section by providing an explicit formula for $\#\operatorname{SubMon}(G\times[n])$ when $G$ is a finite Abelian group.

Theorem 2.11.

Suppose $M=G$ is a finite Abelian group and $n$ is a natural number. Then

[TABLE]

where $c_{m}$ is the number of length $m$ chains in the subgroup lattice of $G$ and $h$ is the height of $\operatorname{Sub}(G)$ , i.e., the length of the longest chain in the subgroup lattice.

Proof.

First note that the asymptotic formula follows immediately from the explicit formula.

We now determine the entries of $W(G)$ . Observe that every submonoid of $G$ is in fact a subgroup of $G$ (since $G$ is a finite group). Further note that every finite Abelian group $A$ has exactly two ideals, itself and the empty set. Based on these facts, we can deduce that for $A,B\in\operatorname{SubMon}(G)$ ,

[TABLE]

Let $k=\#\operatorname{Sub}(G)$ be the number of subgroups of $G$ . Then $W=W(G)$ is a $k\times k$ lower triangular matrix with $2$ s on the diagonal and lower triangle entries either [math] or $1$ with $W_{A,B}=1$ if and only if $B\subsetneq A$ . This allows us to write

[TABLE]

for $L$ a strictly-lower triangular matrix of [math]s and $1$ s with $L_{A,B}=1$ if and only if $B\subsetneq A$ .

By the binomial theorem, which can be used since $2I_{k}$ and $L$ commute,

[TABLE]

Let $\mathbf{1}$ be the $k\times 1$ column vector of all $1$ s. Then by Section 2.1,

[TABLE]

Now observe that $L$ is the adjacency matrix for the directed graph on $\operatorname{Sub}(G)$ with a directed edge from $A$ to $B$ if and only if $B$ is a proper subgroup of $A$ . By the usual interpretation of $L^{m}$ in terms of length $m$ walks, we see that $\mathbf{1}^{\top}L^{m}\mathbf{1}=c_{m}$ the number of length $m$ chains in $\operatorname{Sub}(G)$ . This demonstrates that

[TABLE]

as desired. ∎

2.2. Ordinary generating function for $\operatorname{SubMon}(M\times[n])$

Suppose $M$ is a finite commutative monoid and let $W=W(M)$ . We now use the results of [Sta11, Chapter 4] to deduce some results about generating functions.

Let $F_{ij}(x)$ denote the ordinary generating of the $(i,j)$ -entry of $W^{n}$ :

[TABLE]

By [Sta11, Theorem 4.7.2], we have

[TABLE]

where $(B:j,i)$ denotes the matrix obtained by removing the $j$ -th row and $i$ -th column of $B$ .

Now let $F(x)$ be the ordinary generating function for $\#\operatorname{SubMon}(M\times[n])$ . By Section 2.1,

[TABLE]

so we get the following result.

Proposition 2.12.

Suppose $M$ is a finite commutative monoid and

[TABLE]

Then

[TABLE]

where $W=W(M)$ and $w(A,A)$ is the number of ideals of the submonoid $A\leq M$ .

Proof.

From the preceding discussion, the only thing left to check is the expression for the denominator of $F(x)$ . This follows because $W$ is lower triangular with eigenvalues $w(A,A)$ , $A\leq M$ . ∎

3. Enumerating submonoids of join-semilattices

3.1. General results

We now specialize to the case where $M=P$ is a finite join-semilattice (i.e., commutative idempotent monoid; see Section 2(b)). In Section 3.1, we prove that $W=W(P)$ is diagonalizable, and this will give us excellent control over $\#\operatorname{SubMon}(P\times[n])$ .

Fix a linear extension of $\operatorname{SubMon}P$ so that $W(P)$ is lower triangular (see Section 2.1). The following lemma is a basic fact from linear algebra. We first encountered it at [hu].

Lemma 3.1.

Suppose $W$ is a triangular square matrix (lower or upper) with entries in a field $F$ , and suppose that the diagonal entries of $W$ are arranged so that if two agree, then all diagonal entries between them agree. Then $W$ is diagonalizable over $F$ if and only if each square diagonal block with a single value on its diagonal is in fact a diagonal matrix as depicted in Figure 2. ∎

To prove the diagonalizability of $W(P)$ , we will show that if two distinct submonoids $A,B\leq P$ satisfy $w(A,A)=w(B,B)$ , then $A$ and $B$ are incomparable, whence $w(A,B)=0$ . For the purposes of this proof, recall that an antichain in a poset is a collection of mutually incomparable elements; it is a basic fact from order theory that antichains are in bijection with ideals. (The bijection takes an antichain to the ideal generated by it, which is given by its upward closure, and it takes an ideal to its set of minimal elements.)

Lemma 3.2.

Suppose $P$ is a finite join-semilattice. Then the map

[TABLE]

is strictly increasing.

Proof.

Suppose $A<B\leq P$ . We have

[TABLE]

Antichains in $A$ are also antichains in $B$ , so $w(A,A)\leq w(B,B)$ . Since $A<B$ , we may take $x\in B\smallsetminus A$ to exhibit an antichain $\{x\}$ in $B$ that is not an antichain in $A$ . Thus $w(A,A)<w(B,B)$ , as required. ∎

Before moving on to diagonalizability, we use Section 3.1 to pin down the asymptotics of $\#\operatorname{SubMon}(P\times[n])$ :

Theorem 3.3.

Suppose $P$ is a finite join-semilattice. Then

[TABLE]

where $C=C(P)$ is a positive constant and $i(P)=w(P,P)$ is the number of antichains of $P$ .

Proof.

By Section 2.1, it suffices to show that the multiplicity of the eigenvalue $i(P)=w(P,P)$ is one. All eigenvalues are of the form $w(A,A)$ , and Section 3.1 implies that $A\mapsto w(A,A)$ achieves its maximum value $w(P,P)$ once (because $P$ is the unique maximum of $\operatorname{SubMon}P$ ). ∎

Theorem 3.4.

Let $P$ be a finite join-semilattice. Then $W(P)$ is diagonalizable with eignevalues the number of antichains in $A$ , $A\in\operatorname{SubMon}P$ .

Proof.

This follows directly from Section 3.1 and Section 3.1. ∎

The diagonalizability of $W(P)$ has immediate enumerative consequences:

Corollary 3.5.

Suppose $P$ is a finite join-semilattice and let $\Lambda=\Lambda(P)$ be the finite set of positive integers

[TABLE]

For $\lambda\in\Lambda$ , let $b_{\lambda}=b_{\lambda}(P)$ be the rational number

[TABLE]

where $Q_{\lambda}=Q_{\lambda}(P)$ is the matrix acting by projection onto the $\lambda$ -eigenspace of $W(P)$ when considering the decomposition into eigenspaces. Then

[TABLE]

Proof.

This follows from Section 2.1 and Section 3.1 via the usual formulæ for powers of diagonalizable matrices. The formula for the $b_{\lambda}$ follows from the spectral decomposition of $W(P)$ . ∎

*Remark 3.6**.*

While the natural numbers in $\Lambda(P)$ encode combinatorial structure as the number of antichains in submonoids of $P$ , the rational numbers $b_{\lambda}$ have resisted combinatorial interpretation thus far. The subsequent formulæ we will derive for specific examples demonstrate that these numbers tend to have large denominators, and we view it as an important open question to compute them — or, equivalently, an eigenbasis of $W(P)$ — in terms of $P$ . Some explicit computations are included in Appendix A.

The decomposition

[TABLE]

for the spectral projectors suggests instead attempting to interpret the normalized coefficients $b_{\lambda}\prod_{\mu\neq\lambda}(\lambda-\mu)$ , which are always integers. Despite numerous searches on the OEIS, these values remain similarly mysterious. That said, see Section 3.1 for a method to determine the $b_{\lambda}$ in terms of initial values and a Vandermonde matrix.

Section 3.1 implies that $\#\operatorname{SubMon}(P\times[n])$ is a constant-recursive sequence. The following result is standard for such sequences; see [Sta11, Theorem 4.1.1].

Proposition 3.7.

Let $P$ be a finite join-semilattice, and let $a_{i}$ be the coefficient of the degree $i$ term in $\prod_{\lambda\in\Lambda}(1-\lambda x)$ . Then for $n\geq\#\Lambda$ ,

[TABLE]

Proof.

As before, write $F(x)$ for the ordinary generating function of $\#\operatorname{SubMon}(P\times[n])$ . By Section 3.1, we can also express $F(x)$ as

[TABLE]

where $f(x)$ is a polynomial with rational coefficients of degree $\#\Lambda-1$ . Thus

[TABLE]

and matching the terms of degree $n\geq\#\Lambda$ on both sides gives the recurrence. ∎

Finally, we observe that the $b_{\lambda}(P)$ coefficients of Section 3.1 can be derived from $\Lambda(P)$ and the initial $\#\Lambda(P)$ terms of $\#\operatorname{SubMon}(P\times[n])$ .

Proposition 3.8.

Suppose $P$ is a finite join-semilattice, let $\Lambda=\Lambda(P)=\{\lambda_{1},\ldots,\lambda_{k}\}$ with cardinality $k$ , and let $S_{n}=\#\operatorname{SubMon}(P\times[n])$ . Let $V$ be the $k\times k$ Vandermonde matrix

[TABLE]

Then $b_{i}:=b_{\lambda_{i}}(P)$ are the unique rational numbers satisfying

[TABLE]

Proof.

By Section 3.1, the exponential generation function $S(z)$ of $S_{n}$ takes the form

[TABLE]

Thus

[TABLE]

The system of equations given by the right-hand equality for $0\leq r\leq k-1$ is exactly $Vb=S$ for $b=(b_{1},\ldots,b_{k})^{\top}$ , $S=(S_{0},\ldots,S_{k-1})^{\top}$ . This system has a unique solution because $V$ is invertible. (Indeed, it is a standard fact that $\det V=\prod_{1\leq i<j\leq k}(\lambda_{j}-\lambda_{i})$ , which is nonzero since the $\lambda_{i}$ are distinct.) ∎

3.2. Formulæ for specific semilattices

For a general finite join-semilattice $P$ , the problem of determining $\Lambda(P)$ and the associated $b_{\lambda}$ seems to be quite hard. Given the relation between $\Lambda(P)$ and the enumeration of antichains, this should come as no surprise. (See Section 3.3 for additional comments on the inherent difficulty of this problem.)

Our first result in this direction is a complete determination of $\Lambda([m])$ .

Proposition 3.9.

For a nonnegative integer $m$ , the eigenvalues of $W([m])$ are

[TABLE]

Proof.

The submonoids of $[m]$ are exactly the subsets $S\subseteq[m]$ that contain [math]. Furthermore, $w(S,S)=\#S+1$ since an antichain in a linearly ordered set is either a singleton or empty. ∎

We now consider the join-semilattice $[m]\times[1]$ .

Proposition 3.10.

For $m$ a nonnegative integer, the eigenvalues of $W([m]\times[1])$ are

[TABLE]

Proof sketch.

We leave a number of details to the reader, but the idea is as follows.

First note it is always the case that the elements of $\Lambda(P)$ are bounded below by $2$ and above by $w(P,P)$ , which is equal to $\binom{m+3}{2}$ in this case.

The reader may check that the cardinality function is a grading on the poset of submonoids. Thus to show that $\binom{m+3}{2}-1$ is not in $\Lambda([m]\times[1])$ , it suffices to show that each submonoid of cardinality $2m+1$ has at most $\binom{m+3}{2}-2$ antichains. Such submonoids do not contain either $(0,1)$ or $(i,0)$ for some $i>0$ . In the former case, the submonoid does not have antichains $\{(0,1)\}$ and $\{(0,1),(1,0)\}$ ; in the latter, it does not have antichains $\{(i,0)\}$ and $\{(0,1),(i,0)\}$ . We conclude that $\binom{m+3}{2}-1$ is not an eigenvalue.

Finally, the reader may perform a case-by-case check that the other potential eigenvalues $\lambda$ , $2\leq\lambda\leq\binom{m+3}{2}-2$ are obtained. ∎

*Remark 3.11**.*

We have not found a general formula for $b_{\lambda}([m]\times[1])$ , but we produce some specific values in Appendix A.

Fix $k\geq 1$ . We now consider the lattice $M_{k}=\{\bot,1,2,\ldots,k,\top\}$ in which $\bot$ is the minimum element, $\top$ is the maximum, and $1,2,\ldots,k$ are incomparable to each other. This lattice is the $k$ -fold parallel composition of $[2]$ with itself.

Proposition 3.12.

For $k$ a positive integer, the eigenvalues of $W(M_{k})$ are

[TABLE]

Proof sketch.

The submonoids of $M_{k}$ are of the following forms:

$\cdot$

Type A submonoids are of the form $A_{S}=S\cup\{\bot,\top\}$ for $S\subseteq\{1,2,\ldots,k\}$ .

$\cdot$

Type B submonoids are of the form $B_{i}=\{\bot,i\}$ for $1\leq i\leq k$ .

$\cdot$

The only Type C submonoid is the trivial submonoid $C$ .

One then checks the following computations by inspection:

[TABLE]

Furthermore, for any other choices of $A,B\leq M_{k}$ , $w(A,B)=0$ . This amounts to a complete computation of $W(M)$ and in particular gives the desired diagonal entries. ∎

Note that Figure 1 gives the weighted graph and matrix in the case $k=2$ .

In order to explore the formula of Section 3.1 we produced a short Sage Math script that computes $W(P)$ , its diagonalization, and an eigenbasis. We have recorded some results in Appendix A.

3.3. Max-closed relations and poly-Bernoulli numbers

In [Knu24], Don Knuth recounts the combinatorics of poly-Bernoulli numbers in terms of parades, acyclic bipartite orientations, doubly bounded permutations, lonesum matrices, and other structures. One may view the max-closed relations he considers in [Knu24, §14] as subsemigroups of $[m-1]\times[n-1]$ . Since there are twice as many subsemigroups as submonoids (with two-to-one mapping given by $(-)\cup\bot$ ), we deduce that

[TABLE]

where $B_{m,n}$ is the $(m,n)$ -th poly-Bernoulli number [Kan97], with $B_{m,n}=B_{n}^{(-m)}$ and

[TABLE]

Theorem 3.13 ([Kan97, Knu24]).

The poly-Bernoulli numbers may be expressed as

[TABLE]

Their exponential generating function is given by

[TABLE]

∎

From this, we deduce the following in the notation of Section 3.1.

Theorem 3.14.

For $m\geq 0$ and $2\leq j\leq m+2$ ,

[TABLE]

∎

*Remark 3.15**.*

The above formula was independently deduced in [HMOO22, Theorem 4.7] using the language of saturated transfer systems (see Section 4).

We now give an independent computation of these coefficients using the technology of Section 3.1. The important point is that, once the matrix $W$ is identified, the problem is completely reduced to computations of linear algebra.

Proof of Section 3.3.

The submonoids $A\leq[m]$ are exactly the subsets of $[m]$ containing [math]; the non-empty ideals of $A$ are in bijection with the elements of $A$ itself, via sending each ideal to its least element. Let $A,B\leq[m]$ . By inspection we thus see that

[TABLE]

These entries define the matrix $W=W([m])$ .

We claim that the matrix

[TABLE]

is an eigenmatrix for $W$ , satisfying that $Q^{-1}WQ$ is diagonal. We prove this by showing by induction on $\#A-\#B$ that $(WQ)_{A,B}=(\#B+1)Q_{A,B}$ for all $B\subseteq A$ . The base case corresponds to $B=A$ and follows directly from the definitions.

Given $B\subsetneq A$ , let $a_{0}=\min(A\smallsetminus B)$ , and let $r=1+\#\{b\in B:b\leq a_{0}\}$ . Then for $B\subseteq C\subseteq A\smallsetminus\{a_{0}\}$ , we have

[TABLE]

Splitting the sum $(WQ)_{A,B}=\sum_{B\subseteq C\subseteq A}W_{A,C}Q_{C,B}$ into the cases $a_{0}\in C$ and $a_{0}\notin C$ , applying the above computations, and using the inductive hypothesis gives the inductive step. Thus $Q$ is an eigenmatrix.

We now wish to compute $Q^{-1}\mathbf{1}$ and $\mathbf{1}^{T}Q$ . We first claim that $(Q^{-1}\mathbf{1})_{A}=\frac{1}{2}(\#A+1)!$ . This follows from the fact that $\sum_{B\subseteq A}\frac{1}{2}(\#B+1)!Q_{A,B}=1$ . The latter is proved by induction on $\#A$ , by splitting the sum according to whether $B$ contains $\max(A)$ or not.

Let $r_{i}$ denote the size of the “ $i$ -th run” in $[m]\smallsetminus B$ . Explicitly, this means that, if $B=\{0=b_{1}<\dots<b_{\#B}\}$ and $b_{\#B+1}=m+1$ , we let $r_{i}=b_{i+1}-b_{i}-1$ . Then $r_{i}$ is the number of choices of $x\in[m]\smallsetminus B$ so that $1+\#\{b\in B:b\leq x\}=i$ . Thus

[TABLE]

Now,

[TABLE]

We conclude Section 3.3 thanks to the standard identity

[TABLE]

see for instance [OLBC10, Equation 26.8.5]. ∎

We now consider the multivariable exponential generating function for the number of submonoids of $[\ell]\times[m]\times[n]$ . We define

[TABLE]

and also set

[TABLE]

where $t=e^{x},u=e^{y},v=e^{z}$ . Because $\#\operatorname{SubMon}([\ell-1]\times[m-1])$ is equal to the poly-Bernoulli number $B_{\ell,m}$ , we have

[TABLE]

By Section 3.1, we also have

[TABLE]

This partial information is consistent with the following conjecture.

Conjecture 3.16.

The generating function $G(x,y,z)$ is a rational function in $e^{x},e^{y},e^{z}$ , as is the higher order multivariable version for $\#\operatorname{SubMon}(\prod_{i}[n_{i}-1])$ .

An explicit formula for $F(t,u,v)$ (and hence, for $G(x,y,z)$ ) and its higher order variants would give a very satisfying answer to the most general version of [Knu24, Problem 10].

*Remark 3.17**.*

The transfer matrix method seems to fall short in enumerating submonoids of $\prod[n_{i}]$ because it does not sufficiently account for symmetry. We also note that [BHK*+*25, Remark 2.10] establishes that submonoids of $[1]^{n}$ are counted by OEIS entry A102896, which is only known for $n\leq 7$ , thus suggesting the difficulty inherent in establishing base cases in an inductive approach to enumerating $\operatorname{SubMon}\prod[n_{i}]$ .

4. Connection to equivariant stable homotopy theory

In this section we elaborate on the connection between submonoids of a semilattice and saturated transfer systems, which are important objects of study in equivariant stable homotopy theory, as they are closely related to equivariant operads. For more context, we refer the reader to [HMOO22, Rub21, BHO*+*24].

Definition 4.1.

Let $(P,\leq)$ be a finite lattice. A saturated transfer system on $P$ is a partial order $R$ on $P$ satisfying the following conditions:

(1)

$R$ refines $\leq$ , i.e., if $xRy$ , then $x\leq y$ ; 2. (2)

$R$ is closed under restriction, i.e., if $xRz$ and $y\in P$ , then $(x\wedge y)R(z\wedge y)$ ; 3. (3)

$R$ is decomposable, i.e., if $xRz$ and $x\leq y\leq z$ , then $xRy$ and $xRz$ .

We denote by $\operatorname{SatTr}(P)$ the set of saturated transfer systems of $P$ . This is a poset by refinement (i.e., inclusion).

Condition (3) is called the saturation condition, a partial order that satisfies (1) and (2) is called a transfer system.

We can consider a saturated transfer system in terms of its Hasse diagram, and as such, we can consider the connected components. Proposition 2.1 of [BHK*+*25] shows that each connected component has a unique minimal element. The following result combines [BHK*+*25, Theorem 2.8] and [BCH*+*25, Theorem 4.5].

Theorem 4.2.

There is an order reversing bijection

[TABLE]

given by taking a saturated transfer system $R$ to the set minimal elements of the connected components with respect to $R$ .

In fact, [BHK*+*25] shows an order reversing bijection between $\operatorname{SatTr}(P)$ and $\operatorname{End}^{\circ}(P)$ , where the latter denotes the set of interior operators of $P$ , that is order preserving maps that are idempotent (for all $x\in P$ , $f(f(x))=f(x)$ ) and contractive (for all $x\in P$ , $f(x)\leq x$ ). This is a poset using the pointwise partial order, where $f\leq g$ is and only if $f(x)\leq g(x)$ for all $x\in P$ . The result in [BCH*+*25] connects $\operatorname{End}^{\circ}(P)$ with $\operatorname{SubMon}(P,\vee)$ .

Via this bijection, we can translate each of the enumerative and asymptotic results of Section 3 to give analogous results for the count of saturated transfer systems on $P\times[n]$ . In the rest of this section, we show how to obtain the transition matrix directly from the data of saturated transfer systems. We present independent proofs of the enumeration techniques, as these ideas might be of interest to those studying transfer systems.

*Remark 4.3**.*

The authors actually discovered the results for transfer systems on finite lattices first, and only later realized that they could be generalized to submonoids of finite commutative monoids. We chose to emphasize submonoids in our treatment since they are both more general and more familiar.

4.1. Saturated transfer systems on $P\times[n]$

Let $P$ be a finite lattice. We are interested in saturated transfer systems $R$ on $P\times[n]$ . It will be convenient to have some language to discuss a given system of saturated covers on this lattice. The below definition is illustrated in Figure 3.

Definition 4.4.

Given a saturated transfer system $S$ on $P\times[n]$ , for $i=0,\dots,n$ , we define the $i$ -th layer, denoted by $S_{i}$ , as the restriction of $S$ to $P\times\{i\}$ . We also write $S_{i}$ for the corresponding partial order on $P$ obtained via the isomorphism $P\times\{i\}\cong P$ . Note that $S_{i}$ is indeed a saturated transfer system on $P$ . For $i=1,\dots,n$ , we write $S^{i}$ for the restriction of $S$ to the height one cylinder $P\times\{i-1<i\}$ . Note that since $P\times\{i-1<i\}$ is isomorphic to $P\times[1]$ , we can think of $S^{i}$ as a saturated transfer system on $P\times[1]$ , with bottom and top layers given by $S_{i-1}$ and $S_{i}$ , respectively.

We start with an observation about saturated transfers systems on $P\times[1]$ , which follows from the fact that the restriction of $(x,1)\to(y,1)$ along $(y,0)$ is $(x,0)\to(y,0)$ .

Lemma 4.5.

If $S$ is a saturated transfer system on $P\times[1]$ , then the layers satisfy $S_{1}\leq S_{0}$ . ∎

For a general saturated transfer system on $P\times[n]$ , it turns out that the information about height one cylinders is all that is needed to recover the entire system.

Lemma 4.6.

Let $(P,\leq)$ be a finite lattice, and $S$ a saturated transfer system on $P\times[n]$ . Then $S$ is completely determined by the tuple $(S^{1},\dots,S^{n})$ of saturated transfer systems on $P\times[1]$ .

Proof.

Given a saturated transfer system $S$ , we consider the relation $Q$ on $P\times[n]$ given by taking $\bigcup_{i=1}^{n}S^{i}$ and then closing under transitivity. We claim $Q=S$ , and thus, we can uniquely recover $S$ from its height one cylinders.

Note that since $S$ contains $\bigcup_{i-1}^{n}S^{i}$ and it is closed under transitivity, then $Q\subseteq S$ . On the other hand, assume $(x,i)S(y,j)$ , for some $x\leq y$ in $P$ , and $0\leq i\leq j\leq n$ . Since $S$ is a saturated transfer system, it is decomposable, and thus we have $(x,i)S(y,i)$ , and $(y,k-1)S(y,k)$ for all $k=i+1,\dots,j$ . All these relations are in $\bigcup_{i-1}^{n}S^{i}$ , and thus $(x,i)Q(y,j)$ , proving that $S\subseteq Q$ . ∎

Following the same strategy as in Section 2, we construct a transition matrix that will aid in the enumeration of saturated transfer systems on $P\times[n]$ . We remark that in this case the proofs are easier, in light of Section 4.1.

Definition 4.7.

Let $(P,\leq)$ be a finite lattice. The saturated transfer system graph of $P$ is the weighted directed graph $ST(P)$ with vertices the saturated transfer systems of $P$ and weight function $w\colon\operatorname{SatTr}(P)\times\operatorname{SatTr}(P)\to\mathbb{N}$ given by

[TABLE]

After choosing a linear ordering on $\operatorname{SatTr}(P)$ , we write $W=W(P)$ for the associated adjacency matrix with $W_{R,Q}=w(R,Q)$ .

*Remark 4.8**.*

The result of Section 4.1 implies that if $w(R,Q)\neq 0$ , then $R\leq Q$ in the refinement order of $\operatorname{SatTr}(P)$ . The converse is also true, as if $R\leq Q$ , we can always construct the transfer system on $P\times[1]$ given by placing $Q$ on the $P\times\{0\}$ layer, $R$ on the $P\times\{1\}$ layer, and including no vertical relations.

Thus we can think of the graph as enhancing the directed graph given by the poset $(\operatorname{SatTr}(P),\leq)$ with positive weights.

Proposition 4.9.

Let $P$ be a finite lattice with weighted adjacency matrix $W=W(P)$ . Then for all $R,Q\in\operatorname{SatTr}(P)$ , and all $n\geq 0$ ,

[TABLE]

Thus,

[TABLE]

Proof.

The second equality follows directly from the first. To prove the first assertion, note that Section 4.1 implies that a saturated transfer system is determined by a path $S_{n}\leq S_{n-1}\leq\dots\leq S_{0}$ in the poset $\operatorname{SatTr}(P)$ , and for each such path, a choice of how to fill the cylinders $S^{i}$ , with the condition that the bottom and top layers of $S^{i}$ are $S_{i-1}$ and $S_{i}$ , respectively, which is precisely being counted by $W_{S_{i},S_{i-1}}$ . The entries of the powers of the adjacency matrix are precisely giving the count when $S_{0}$ and $S_{n}$ are fixed. ∎

We finish this section by proving the following result. Recall the bijection $\chi$ of Section 4, and the weighted graphs $G(P)$ of Section 2.1 and $ST(P)$ of Section 4.1.

Theorem 4.10.

For a finite lattice $P$ , the bijection $\chi$ induces an isomorphism of weighted graphs between $ST(P)$ and $G(P)$ .

Proof.

Since the vertices of $ST(P)$ are the saturated transfer systems of $P$ and the vertices of $G(P)$ are the submonoids of $(P,\vee)$ , Section 4 establishes a bijection between both sets of vertices. It thus remains to show that for saturated transfer systems $R,Q$ of $P$ ,

[TABLE]

Recall that the left-hand side is counting the saturated transfer systems on $P\times[1]$ with bottom layer given by $Q$ and top layer given by $R$ , while the right-hand side is counting the ideals (i.e., upward-closed sets) of $\chi(R)$ whose union with $\chi(Q)$ is equal to $\chi(R)$ . We thus construct an explicit bijection between these sets.

To go forward, let $S$ be a saturated transfer system on $P\times[1]$ such that $S_{0}=Q$ and $S_{1}=R$ . Let

[TABLE]

that is, the set of elements in $P$ that correspond to vertical arrows in $S$ . Now consider the set $I=\chi(R)\cap(P\smallsetminus V)$ . The restriction condition on $S$ implies $V$ is downward-closed, and hence $I$ is an upward-closed subset of $\chi(R)$ . We claim that $I\cup\chi(Q)=\chi(R)$ . Note that since $S$ exists, $R\leq Q$ , and since $\chi$ is order reversing, we have that $\chi(Q)\subseteq\chi(R)$ , giving one containment. For the reverse containment, suppose $x\in\chi(R)$ but $x\not\in I$ . This means by definition that $x\in V$ , so $(x,0)S(x,1)$ . Since $x\in\chi(R)$ , $x$ is minimal in its connected component in $R$ . We now show that $x$ is also minimal in its connected component in $Q$ , thus showing that $x\in\chi(Q)$ , and proving the containment. Let $y\leq x$ in $P$ , and suppose for the sake of contradiction that $y\to x$ is in $Q$ , meaning that the following are all in $S$ .

$(x,0)$$(x,1)$$(y,0)$

Then we have that $(y,0)\to(x,0)$ and $(x,0)\to(x,1)$ are in $S$ , which implies by transitivity that $(y,0)\to(x,1)$ is in $S$ . By decomposability, this in turn implies that $(y,1)\to(x,1)$ , which contradicts the minimality of $x$ within its connected component in $R$ , as depicted below.

$(x,0)$$(y,0)$$(x,1)$$(y,1)$

We can reverse the construction as follows. Given an upward-closed set $I\subseteq\chi(R)$ such that $I\cup\chi(Q)=\chi(R)$ , let $\widetilde{V}$ denote the subset of $P$ consisting of those elements whose connected component $C$ in $R$ satisfies that $C\cap(P\smallsetminus I)$ is non-empty. Now construct the relation $S$ on $P\times[n]$ as the closure under transitivity of

[TABLE]

We claim that $S$ is a saturated transfer system on $P\times[1]$ with top and bottom layers given by $R$ and $Q$ , respectively. The latter follows directly from the construction, since closing under transitivity will only add arrows that connect the bottom layer with the top layer.

To prove $S$ is closed under restriction it is enough to prove that $T$ itself is as well, as this property is respected by closing under transitivity (see [Rub21, Theorem A.2]). We split into cases. Since $Q$ is itself closed under restriction, the restriction of an arrow in $Q\times\{0\}$ is in $Q\times\{0\}$ . The same is true for arrows in $R\times\{1\}$ when restricting along elements of the form $(y,1)$ . The restriction of $(x,1)\to(z,1)$ along $(y,0)$ is the arrow $(x\wedge y,0)\to(z\wedge y,0)$ . Now, if the former is in $R\times\{1\}$ , we have that $(x\wedge y,1)\to(z\wedge y,1)$ is also in $R\times\{1\}$ because $R$ is closed under restriction, and from there we get that $(x\wedge y,0)\to(z\wedge y,0)$ is in $Q\times\{0\}$ , since $R\leq Q$ .

Finally, we consider arrows in $\widetilde{V}\times[1]$ . Showing that this set is closed under restriction is equivalent to showing that $\widetilde{V}$ is downward-closed. Let $x\in\widetilde{V}$ and $y\leq x$ . Then there exists $z\in P\smallsetminus I$ such that $x$ and $z$ are in the same connected component with respect to $R$ , meaning that there is a zig-zag of arrows in $R$ connecting $x$ and $z$ . Taking the restriction of this zig-zag with respect to $y$ , we obtain a zig-zag of arrows in $R$ connecting $y=x\wedge y$ with $z\wedge y$ . Now, since $I$ is upward-closed we have that $P\smallsetminus I$ is downward-closed, and hence, $z\wedge y$ is in $P\smallsetminus I$ . This implies that $y\in\widetilde{V}$ , as wanted.

It remains to show that $S$ is decomposable. Note that $T$ is decomposable, since $R$ and $Q$ are as well, and the arrows in $\widetilde{V}\times[1]$ are not composites. The only new arrows that are in $S$ but not in $T$ are of the form $(x,0)\to(y,1)$ with $x<y$ . Since $R$ and $Q$ are closed under transitivity, if such an arrow is in $S$ , it is because there exists $z\in\widetilde{V}$ , such that $xQz$ and $zRy$ , as below.

$(y,1)$$(y,0)$$(x,0)$$(x,1)$$(z,0)$$(z,1)$

Thus $y$ and $z$ are in the same connected component with respect to $R$ , and hence $y\in\widetilde{V}$ .

Let $w$ be the minimal element in the connected component of $z$ with respect to $R$ , as depicted by the top red arrow in the diagram below.

$(y,1)$$(y,0)$$(x,0)$$(x,1)$$(z,0)$$(z,1)$$(w,1)$$(w,0)$

Note that $w\in P\smallsetminus I$ , since $P\smallsetminus I$ is downward-closed and the intersection of the connected component of $z$ with $P\smallsetminus I$ is non-empty. Given that $w\in\chi(R)=I\cup\chi(Q)$ , we then must have that $w\in\chi(Q)$ . Thus $w$ is the minimal element in its connected component with respect to $Q$ , and since $R\leq Q$ , we have the red arrow at the bottom, implying that this is the connected component of $z$ . Given that $xQz$ , this is also the connected component of $x$ , and thus we have $w\leq x\leq z$ . Since $wRz$ , decomposability of $R$ implies that $xRz$ , and hence $xRy$ .

Any decomposition of $(x,0)\to(y,1)$ is of the form $(x,0)\to(u,0)\to(y,1)$ or $(x,0)\to(u,1)\to(y,1)$ for some $u$ such that $x\leq u\leq y$ . Since $xRy$ and $R$ is decomposable, we have $xRu$ and $uRy$ , meaning that $(x,0)\to(u,0)$ and $(u,1)\to(y,1)$ are both in $S$ . Further, since $\widetilde{V}$ is downward-closed, we also have that $(u,0)\to(u,1)$ is in $S$ . Closure under transitivity implies then that both of the components of either our decompositions are in $S$ , proving decomposability.

We leave it to the reader to prove that these two constructions are inverses of each other. ∎

Appendix A Explicit computations

The following table records the output of a short Sage Math script that, for $P$ a finite join-semilattice, computes $W(P)$ , its diagonalization, and an eigenbasis. This allows us to determine the eigenvalues $\Lambda=\Lambda(P)$ and coefficients $b_{\lambda}\in\mathbb{Q}$ , $\lambda\in\Lambda$ of Section 3.1, giving

[TABLE]

The table displays the finite join-semilattice $P$ , eigenvalues $\lambda\in\Lambda(P)$ , coefficients $b_{\lambda}\in\mathbb{Q}$ , and normalized coefficients $\hat{b}_{\lambda}:=b_{\lambda}\prod_{\mu\neq\lambda\in\Lambda}(\lambda-\mu)\in\mathbb{Z}$ (see Section 3.1).

We note that our computations reveal that $4\in\Lambda(M_{3})$ but $b_{4}(M_{3})=0$ . This is the only example we are aware of for which $b_{\lambda}=0$ .

Bibliography13

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TL;DR

Contribution

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Abstract

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Taxonomy

Enumerating submonoids of finite commutative monoids

Abstract.

Contents

1. Introduction

Question**.**

Theorem** (Section 2.1).**

Theorem** (Section 2.1).**

Theorem** (Section 3.1, Section 3.1).**

Remark 1.1*.*

Overview

Acknowledgments

2. Transition matrices for submonoids

Example 2.1**.**

Definition 2.2**.**

2.1. The submonoid graph

Definition 2.3**.**

Remark 2.4*.*

Definition 2.5**.**

Example 2.6**.**

Lemma 2.7**.**

Proof.

Theorem 2.8**.**

Lemma 2.9**.**

Proof.

Theorem 2.10**.**

Proof.

Theorem 2.11**.**

Proof.

2.2. Ordinary generating function for SubMon⁡(M×[n])\operatorname{SubMon}(M\times[n])SubMon(M×[n])

Proposition 2.12**.**

Proof.

3. Enumerating submonoids of join-semilattices

3.1. General results

Lemma 3.1**.**

Lemma 3.2**.**

Proof.

Theorem 3.3**.**

Proof.

Theorem 3.4**.**

Proof.

Corollary 3.5**.**

Proof.

Remark 3.6*.*

Proposition 3.7**.**

Proof.

Proposition 3.8**.**

Proof.

3.2. Formulæ for specific semilattices

Proposition 3.9**.**

Proof.

Proposition 3.10**.**

Proof sketch.

Remark 3.11*.*

Proposition 3.12**.**

Proof sketch.

3.3. Max-closed relations and poly-Bernoulli numbers

Theorem 3.13** ([Kan97, Knu24]).**

Theorem 3.14**.**

Remark 3.15*.*

Proof of Section 3.3.

Conjecture 3.16**.**

Remark 3.17*.*

4. Connection to equivariant stable homotopy theory

Definition 4.1**.**

Theorem 4.2**.**

Remark 4.3*.*

4.1. Saturated transfer systems on P×[n]P\times[n]P×[n]

Definition 4.4**.**

Lemma 4.5**.**

Lemma 4.6**.**

Proof.

Definition 4.7**.**

Remark 4.8*.*

Question.

Theorem (Section 2.1).

Theorem (Section 2.1).

Theorem (Section 3.1, Section 3.1).

*Remark 1.1**.*

Example 2.1.

Definition 2.2.

Definition 2.3.

*Remark 2.4**.*

Definition 2.5.

Example 2.6.

Lemma 2.7.

Theorem 2.8.

Lemma 2.9.

Theorem 2.10.

Theorem 2.11.

2.2. Ordinary generating function for $\operatorname{SubMon}(M\times[n])$

Proposition 2.12.

Lemma 3.1.

Lemma 3.2.

Theorem 3.3.

Theorem 3.4.

Corollary 3.5.

*Remark 3.6**.*

Proposition 3.7.

Proposition 3.8.

Proposition 3.9.

Proposition 3.10.

*Remark 3.11**.*

Proposition 3.12.

Theorem 3.13 ([Kan97, Knu24]).

Theorem 3.14.

*Remark 3.15**.*

Conjecture 3.16.

*Remark 3.17**.*

Definition 4.1.

Theorem 4.2.

*Remark 4.3**.*

4.1. Saturated transfer systems on $P\times[n]$

Definition 4.4.

Lemma 4.5.

Lemma 4.6.

Definition 4.7.

*Remark 4.8**.*

Proposition 4.9.

Theorem 4.10.