A note on unitizations of generalized effect algebras

TL;DR

This paper demonstrates that the process of unitization in generalized effect algebras forms a monadic adjunction with the forgetful functor to effect algebras, establishing a categorical foundation for the construction.

Contribution

It proves that the unitization functor is a left adjoint to the forgetful functor and that this adjunction is monadic, clarifying the categorical relationship.

Findings

The forgetful functor is a right adjoint.

The unitization construction is the left adjoint.

The adjunction is monadic.

Abstract

There is a forgetful functor from the category of generalized effect algebras to the category of effect algebras. We prove that this functor is a right adjoint and that the corresponding left adjoint is the well-known unitization construction by Hedl\'ikov\'a and Pulmannov\'a. Moreover, this adjunction is monadic.

Equations

$$\begin{gathered} w(x\oplus y)=w((b{\ominus }_{E}a{)}^{\ast })=((b\ominus a{)}^{\prime },1) \\ =(b^{\prime }\oplus a,1)=(a,0)\oplus (b^{\prime },1)=w(a)\oplus w(b^{\ast })=w(x)\oplus w(y). \end{gathered}$$

$$F\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$UF(P)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bar{s}(x)=\{\begin{array}{ll}s(x)& x\in P\\ 1-s(x)& x\in P^{\ast }\end{array}$$

$$U(A)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{C}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1 Introduction and preliminaries

1.1 Introduction

In hedlikova1996generalized , authors proved that every generalized effect algebra can be embedded into an effect algebra. The construction was subsequently studied and applied by several authors, for example in paseka2009isomorphism , riecanova2005generalized , riecanova2008effect . A generalization of the unitization construction to pseudoeffect algebras was recently introduced and studied in foulis2014unitizing

It is easy to see that this unitization construction is functorial. We prove that this unitization functor is left adjoint to the forgetful functor from the generalized effect algebras to effect algebras and that this adjunction is monadic. Thus, the category of effect algebras is the category of algebras for a monad defined on the category of generalized effect algebras.

We assume working knowledge of basic category theory mac1998categories and theory of effect algebras DvuPul:NTiQS .

1.2 Generalized effect algebras

Let $P$ be a partial algebra with a nullary operation [math] and a binary partial operation $\oplus$. Denote the domain of $\oplus$ by $\perp$. $P$ is called a generalized effect algebra iff for all $a,b,c\in P$ the following conditions are satisfied :

1. (P1)

$a\perp b$ implies $b\perp a$, $a\oplus b=b\oplus a$. 2. (P2)

$b\perp c$ and $a\perp b\oplus c$ implies $a\perp b$, $a\oplus b\perp c$, $a\oplus(b\oplus c)=(a\oplus b)\oplus c$. 3. (P3)

$a\perp 0$ and $a\oplus 0=a$. 4. (P4)

$a\oplus b=a\oplus c$ implies $b=c$. 5. (P5)

$a\oplus b=0$ implies $a=0$.

In a generalized effect algebra $P$, we denote $a\leq b$ iff $a\oplus c=b$ for some $c\in P$. It is easy to see that $\leq$ is a partial order and that [math] is the least element of the poset $(P,\leq)$. We denote $c=a\ominus b$ iff $a=b\oplus c$. Owing to cancellativity, $\ominus$ is a well-defined partial operation with domain $\geq$.

Let $P_{1}$, $P_{2}$ be generalized effect algebras. A map $f:P_{1}\to P_{2}$ is called a morphism of generalized effect algebras if and only if it satisfies the following conditions.

• •

$f(0)=0$.

• •

If $a\perp b$, then $f(a)\perp f(b)$ and $f(a\oplus b)=f(a)\oplus f(b)$.

A morphism is $f:P_{1}\to P_{2}$ is full if $f(a)\perp f(b)$ implies that there are $a_{1},b_{1}\in P_{1}$ such that $a_{1}\perp b_{1}$, $f(a)=f(a_{1})$ and $f(b)=f(b_{1})$.

1.3 Effect algebras

An effect algebra is an generalized effect algebra bounded above. Unwinding this definition, we observe that an effect algebra is partial algebra $(E;\oplus,0,1)$ with a binary partial operation $\oplus$ and two nullary operations $0,1$ such that the reduct $(E;\oplus,0)$ is a generalized effect algebra and $1$ is the greatest element of $E$.

Effect algebras were introduced by Foulis and Bennett in their paper FouBen:EAaUQL . See also KopCho:DP and GiuGre:TaFLfUP for equivalent definitions, introduced independently.

Let $E_{1}$, $E_{2}$ be effect algebras. A map $\phi:E_{1}\to E_{2}$ is called a morphism of effect algebras if and only if it is a morphism of generalized effect algebras satisfying the condition $\phi(1)=1$. A morphism $\phi:E_{1}\to E_{2}$ is a full morphism if and only if $\phi(a)\perp\phi(b)$ implies that there are $x,y\in E_{1}$ such that $\phi(x)=a$, $\phi(y)=b$ and $x\perp y$. A full, bijective morphism is an isomorphism.

2 The unitization functor

The category of generalized effect algebras is denoted by ${\bf GEA}$, the category of effect algebras is denoted by ${\bf EA}$, $U:{\bf EA}\to{\bf GEA}$ is the evident forgetful functor.

Let us define a functor $F:{\bf GEA}\to{\bf EA}$, called unitization.

For a generalized effect algebra $P$, $F(P)$ is a partial algebra with an underlying set $P\dot{\cup}P^{*}$, where $P\cap P^{*}=\emptyset$ and $a\mapsto a^{*}$ is a bijection from $P$ to $P^{*}$, equipped with a partial operation given as follows: for all $a,b\in P$,

• •

$a\perp_{F(P)}b$ iff $a\perp_{P}b$ and then $a\oplus_{F(P)}b=a\oplus_{P}b$,

• •

$a\perp_{F(P)}b^{*}$ iff $a\leq b$ and then $a\oplus_{F(P)}b^{*}=(b\ominus_{P}a)^{*}$,

• •

$a^{*}\perp_{F(P)}b$ iff $a\geq b$ and then $a^{*}\oplus_{F(P)}b=(a\ominus_{P}b)^{*}$,

• •

$a^{*}\not\perp_{F(P)}b^{*}$.

This construction was introduced by Hedlíková and Pulmannová in hedlikova1996generalized . They proved that $(F(P),\oplus,0,0^{*})$ is always an effect algebra. The basic idea of the construction predates effect algebras, see janowitz1968note , mayet1991generalized .

Example 1

Let $P$ be the poset on the left-hand side of Figure 1. There is a unique $\oplus$ partial operation on $P$, making $(P,\oplus,0)$ into a generalized effect algebra: $a\oplus b=b\oplus a=c$, $b\oplus b=d$ and $0\oplus x=x\oplus 0=x$, for all $x\in P$.

The Hasse diagram of the effect algebra $F(P)$ appears on the right-hand side of the picture.

Example 2

Let us consider generalized effect algebra (in fact, a total monoid) $(\mathbb{N},+,0)$, where $+$ is the ordinary addition of natural numbers. Then $F(\mathbb{N})$ is a totally ordered MV-algebra, also known under the name Chang’s MV-algebra.

For a morphism of generalized effect algebras $f:P\to Q$, then $F(f):F(P)\to F(Q)$ is given by $F(f)(a)=a$, $F(f)(a^{*})=(f(a))^{*}$. It is easy to check that $F(f)$ is a morphism in the category ${\bf EA}$ and that $F:{\bf GEA}\to{\bf EA}$ is a functor.

Theorem 2.1

$F$* is left adjoint to $U$.*

Proof

Let us define the unit $\eta$. For every generalized effect algebra $P$, the component $\eta_{P}:P\to UF(P))$ is the embedding $\eta_{P}(x)=x$. This is obviously a natural transformation $1_{{\bf GEA}}\to UF$.

Let $E$ be an effect algebra; to define the component of the counit $\epsilon$ at $E$, we need to take a closer look at $FU(E)$. Let us prove that $w:FU(E)\to E\times\{0,1\}$ given by $w(a)=(a,0)$ and $w(a^{*})=(a^{\prime},1)$, where $a\in E$, is an isomorphism of effect algebras. Indeed, suppose that $x,y\in FU(E)$ and that $x\perp y$. The only nontrivial case we have to check is when there are $a,b\in E$ such that $x=a$, $y=b^{*}$ and $a\leq b$; in this case $x\oplus y=(b\ominus_{E}a)^{*}$ and

[TABLE]

The morphism $w$ is easily seen to be full and bijective, hence an isomorphism.

We may now define $\epsilon_{E}:FU(E)\to E$ as the composition of $w$ with the canonical projection $p:E\times\{0,1\}\to E$. Explicitly, $\epsilon_{E}(x)=x$ for $x\in E$ and $\epsilon_{E}(x^{*})=x^{\prime}$ for $x^{*}\in E^{*}$. The commutativity of the naturality square of $\epsilon$ also clear.

Let us check the triangle identities. We need to prove that, in the categories of endofunctors of ${\bf GEA}$ and ${\bf EA}$, respectively, the triangles

[TABLE]

commute.

To observe the commutativity of the left triangle, let $P$ be a generalized effect algebra. If $x\in P$, then $F(\eta_{P})(x)=x$ and $\epsilon_{F(P)}(x)=x$. If $x^{*}\in P$, then $F(\eta_{P}(x^{*}))=(\eta_{P}(x))^{*}=x^{*}$ and $\epsilon_{F(P)}(x^{*})=x^{*}$.

To observe the commutativity of the right triangle, let $E$ be an effect algebra and let $x\in U(E)$. Then $\eta_{U(E)}(x)=x$ and $U(\epsilon_{FU}(x))=x$.

Let us consider the real interval $[0,1]_{\mathbb{R}}$, equipped with the usual addition of real numbers restricted to $[0,1]_{\mathbb{R}}$, meaning that $a\oplus b$ is defined if and only if $a+b\leq 1$ and then $a\oplus b:=a+b$. A morphism of generalized effect algebras $P\to[0,1]_{\mathbb{R}}$ is called an additive map on $P$.

For an effect algebra $E$, a state on $E$ is an additive map $E\to[0,1]_{\mathbb{R}}$ preserving the unit, so a state is a morphism in ${\bf EA}$.

Corollary 1

Every additive map on $P$ uniquely extends to a state on $F(P)$.

Proof

If $s$ is an additive map, then there is a unique $\bar{s}:F(P)\to[0,1]_{\mathbb{R}}$ such that the diagram

[TABLE]

commutes.

Every state $f$ on an effect algebra must satisfy $f(x^{\prime})=1-f(x)$. Therefore, if $s$ is an additive map on $P$, then the state $\bar{s}$ on $F(P)$ corresponding to $s$ via the bijection established in 1 is necessarily given by

[TABLE]

In a very similar way, one can prove the following:

Corollary 2

There is a natural one-to-one correspondence between ideals of $P$ and morphisms $F(P)\to 2^{2}$, where $2^{2}$ is the Boolean algebra with two atoms.

Lemma 1

The forgetful functor $U:{\bf EA}\to{\bf GEA}$ creates coequalizers.

Proof

Let $f,g\colon A\to B$ be a pair of morphisms in ${\bf EA}$, let $h\colon B\to Z$ be a coequalizer of $f,g$ in ${\bf GEA}$. We need to prove that $h(1)$ is the top element of $Z$. Consider the diagram

[TABLE]

where $t$ is given by $t(x)=h(x)$ and $j$ is the inclusion into $Z$, so that $j\circ t=h$. Since $t\circ f=t\circ g$, there is a unique $u\colon Z\to[0,h(1)]_{Z}$ such that $u\circ h=t$. This gives us $j\circ u\circ h=h$. Since $h$ is an epimorphism, $j\circ u=id_{Z}$ and now we see that for every $x\in Z$, $x=j(u(x))\leq h(1)$, because the range of $j$ is bounded above by $h(1)$.

It remains to prove that $h$ is a coequalizer in ${\bf EA}$. If $E$ is a generalized effect algebra bounded above and $s$ is a top-preserving morphism such that $s\circ f=s\circ g$, then there is a unique ${\bf GEA}$-morphism $u$ such that $u\circ h=s$. However, since both $h$ and $s$ preserve the top element, $u$ must be top-preserving as well.

Recall mac1998categories , that every adjunction

[TABLE]

gives rise to a monad $(UF,\eta,U\epsilon F)$ on $\mathcal{C}$. An adjunction is monadic if $U$ is equivalent to the forgetful functor coming from the category of algebras $\mathcal{C}^{UF}$ for the monad $(UF,\eta,\epsilon)$ and the comparison gives us then an isomorphism $\mathcal{D}\simeq\mathcal{C}^{UF}$.

Theorem 2.2

The adjunction $(F,U,\eta,\epsilon)$ is monadic.

Proof

By Beck’s theorem mac1998categories , an adjunction is monadic if and only if $U$ creates absolute coequalizers. By Lemma 1, $U$ creates all coequalizers.

Corollary 3

${\bf EA}\simeq{\bf GEA}^{UF}$.

Acknowledgements.

This research is supported by grant VEGA G-2/0059/12 of MŠ SR, Slovakia and by the Slovak Research and Development Agency under the contracts APVV-0073-10, APVV-0178-11.