Associative Protomodular Algebras

TL;DR

This paper introduces a new notion of associativity for high-arity operations and proves that a variety of universal algebras contains a group operation if and only if it contains an associative semi-abelian operation.

Contribution

It defines a novel concept of associativity for high-arity operations and establishes a characterization of when a variety includes a group operation.

Findings

A new notion of associativity for high-arity operations is introduced.

A characterization of varieties containing group operations is provided.

The algebraic theory's structure is linked to semi-abelian operations with this new associativity.

Abstract

The notion of associativity (which differs from the straightforward generalization of the usual associativity given by the move of parentheses in the relevant expression) for operations of high arity is introduced. It is proved that the algebraic theory of a variety of universal algebras contains a group operation if and only if it contains a semi-abelian operation which is associative in the sense introduced.

Equations

$$\theta (a_1,a_2,...,a_n,\theta (b_1,b_2,...,b_n,b))=$$

$$=\theta (\theta (a_1,a_2,...,a_n,b_1),\theta (a_1,a_2,...,a_n,b_2),...,\theta (a_1,a_2,...,a_n,b_n),b).$$

$$a\cdot b=\theta (a,a,...,a,b),$$

$$a^{-1}=$$

$$=\theta ({\alpha }_1(e,\theta (a,a,...,a)),{\alpha }_2(\theta (e,\theta (a,a,...,a)),...,{\alpha }_n(e,\theta (a,a,...,a)),a),$$

$${\alpha }_i(a,a)=e_i;$$

$$\theta ({\alpha }_1(a,b),{\alpha }_2(a,b),...,{\alpha }_n(a,b),b)=a.$$

$$\theta (a,b)=ab,$$

$${\alpha }_1(a,b)=a/b,$$

$$e_1=1.$$

$$\theta (a,b)=(a\wedge \rceil b)\vee (b\wedge \rceil a).$$

$$\theta (x,y,z)=(x\vee z)\wedge y,$$

$${\alpha }_1(x,y)=(x\wedge \rceil y),$$

$${\alpha }_2(x,y)=(x\vee \rceil y),$$

$$e_1=0,e_2=1,$$

$$\theta (e_1,e_2,...,e_n,a)=a.$$

$$\theta (a_1,a_2,...,a_n,\theta (b_1,b_2,...,b_n,c))=$$

$$=\theta (\theta (a_1,a_2,...,a_n,b_1),\theta (a_1,a_2,...,a_n,b_2),...,\theta (a_1,a_2,...,a_n,b_n),c).$$

$$\theta (a_1,a_2,...,a_n,b)=a_1{a}_2...a_{n}b$$

$$\theta (a_1,a_2,...,a_n,a_{n+1})=a_i.$$

$$\theta (a_1,a_2,...,a_n,b)=a_{i}b.$$

$$\theta (a_1,a_2,...,a_n,b)=a_1+a_2+...+a_n+b$$

$$a{\circ }_{x}b=\theta (a,x,b).$$

$$a{\circ }_x(b{\circ }_{y}c)=(a{\circ }_{x}b){\circ }_{(a_{{\circ }_x}y)}c,$$

$$\theta (a,b,c)=(a\vee b)\wedge c,$$

$$\theta (a,b,c)=(a\vee c)\wedge b.$$

$$(A,\theta ,({\alpha }_i{)}_{1\le i\le n},(e_i{)}_{1\le i\le n})$$

$$\theta (e_1,e_2,...,e_n,b)=b.$$

$$\theta (a,a,...,a,b)=a,$$

$$(A,\theta ,({\alpha }_i{)}_{1\le i\le n},e)$$

Full text

Key words and phrases: associativity for operations of high arity; protomodular variety; semi-abelian variety.

2000 Mathematics Subject Classification: 08B05, 18C05, 18C10.

1. Introduction

In [7], among many other things P. T. Johnstone and C. M. Pedicchio noted that an algebraic theory (with a constant) contains a group operation if and only if it contains a Mal’cev operation which is associative in the sense of [7]. Note also that there are two important classes of Mal’cev varieties – the classes of protomodular and semi-abelian varieties – which, similarly to the case of Mal’cev varieties, have purely syntactical characterizations due to D. Bourn and G. Janelidze [4]. Recall that the notions of protomodular and semi-abelian varieties were derived from the corresponding categorical notions introduced by D. Bourn [3] and G. Janelidze, L. Marki, and W. Tholen [6], respectively, as abstract settings in which many properties of groups remain valid. The Bourn-Janelidze characterizations require the existence of one operation of arbitrarily high arity (called a protomodular/semi-abelian operation), together with some binary operations and constants that satisfy certain identities (which, in fact, were originally considered by A.Ursini [9], but for different purposes). In view of the above-said a natural question arises whether there is an analog of the Johnstone-Pedicchio statement which would impose an ”associativity-like” condition on a protomodular/semi-abelian operation.

Since the Bourn-Janelidze characterizations enable us to construct explicitly the Mal’cev term in a protomodular/semi-abelian variety, one can attempt to find such an analog just by plugging the explicit form of the Mal’cev term in the associativity identity mentioned in the Johnstone-Pedicchio statement. However, the condition obtained in this way does not provide the basis to be interpreted as a kind of higher associativity, since, for instance, it contains not only the protomodular/semi-abelian operation, but also the binary operations and constants from the Bourn - Janelidze’s result.

In the present paper, we give the analog of the Johnstone-Pedicchio statement whose formulation requires only a kind of higher associativity condition on a semi-abelian term, and does not involve any other operations. To this end, we first introduce the notion of 2-associativity for operations of high arity which is a generalization of the usual associativity condition (but is different from its straightforward generalization given by the move of parentheses in the relevant expression; the term ”1-associativity” is left for the latter generaization). The point here is that the associativity for a binary operation $\times$ on a set $A$ can be formulated as the condition that the mapping $(-\times a)$ from $A$ to the algebra of mappings $Map(A,A)$ (with the composition operation) preserves the operation $\times$, for any $a\in A$. Generalizing this condition to the case of operations $\theta$ of high arity we get the notion of 2-associativity (see Section 3). It is equivalent to the identity

[TABLE]

[TABLE]

An example of such an operation is given by the protomodular operation of the algebraic theory of the variety of Boolean algebras: $\theta(a,b,c)=(a\vee c)\wedge b$. In fact, this operation, like a similar one $\theta(a,b,c)=(a\vee b)\wedge c$, is 2-associative in any distributive lattice.

The main result of the present paper asserts that the algebraic theory of a variety of universal algebras contains a group operation if and only if it contains a 2-associative semi-abelian term $\theta$. In that case the group operation is defined by

[TABLE]

its unit is $e$ and the inverse of an element $a$ is given by

[TABLE]

[TABLE]

where $\alpha_{i}$ are the binary operations and $e$ is the constant from the Bourn-Janelidze characterization of protomodular varieties.

If we look at 2-associative algebras of the simplest semi-abelian variety as high arity analogs of groups, then this result implies that the ”$n$-arity groups” are nothing else but the so-called $n$-enriched groups (in the sense of Section 4 of this paper).

2. Preliminaries

For the definitions of a protomodular category and a semi-abelian category we refer the reader to the papers [3] by D. Bourn and [6] by G. Janelidze, L. Marki, and W. Tholen, respectively.

The characterizations of protomodular and semi-abelian varieties of universal algebras were found by D. Bourn and G. Janelidze in [4]. Namely, they proved that a variety of universal algebras $\mathbb{V}$ is protomodular if and only if its algebraic theory contains, for some natural $n$, constants $e_{1},e_{2},...,e_{n}$, binary operations $\alpha_{1}$, $\alpha_{2}$,…, $\alpha_{n}$ and an $(n+1)$-ary operation $\theta$ such that the following identities are satisfied:

[TABLE]

[TABLE]

$\mathbb{V}$ is semi-abelian if and only if its signature contains a unique constant $e$, and (2.1) and (2.2) are satisfied for $e_{1}=e_{2}=...=e_{n}=e$.

For simplicity, algebras from a protomodular (resp. semi-abelian) variety are called protomodular (resp. semi-abelian). The operation $\theta$ satisfying (2.2) for some $\alpha_{i}$ and $e_{i}$ which in their turn satisfy (2.1) is called protomodular. A protomodular operation is called semi-abelian if all $e_{i}$ are equal.

The motivating example of a semi-abelian variety is the variety of groups; in that case we have

[TABLE]

[TABLE]

[TABLE]

Similarly, any variety whose algebraic theory contains a group operation is protomodular. The varieties of Boolean algebras and Heyting algebras are protomodular too. As is well-known, the algebraic theory of the former variety has a group operation:

[TABLE]

Another protomodular operation of this algebraic theory [2] is given by :

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

For a protomodular operation of the variety of Heyting algebras we refer the reader to [8]; this operation in fact is semi-abelian (nevertherless the variety of Heyting algebras is not semi-abelian!). The same operation makes the variety of Heyting semi-lattices semi-abelian [8]. Other examples of semi-abelian varieties are given by the varieties of loops, left/right semiloops***For loops and left semiloops, the semi-abelian operation and the relevant binary operation are the same as in the case of groups., and locally Boolean distributive lattices [1].

The identities (2.1) and (2.2) imply [2]:

[TABLE]

3. Associativity Conditions for Operations of High Arity

Note that the associativity condition for a binary operation $\theta$ on a set $A$ is equivalent to the condition that the mapping $F:A\rightarrow Map(A,A)$ defined as $F(b)=\theta_{b}=\theta(-,b)$ is a homomorphism (when the set of all mappings $Map(A,A)$ is equipped with the composition operation). Taking this observation into account, below we define the 2-associativity condition for operations of any arities.

Let $A$ be a set equipped with an $(n+1)$-ary operation $\theta$, and let $Map(A^{n},A)$ be the set of all mappings $A^{n}\rightarrow A$. Let us introduce the $(n+1)$-ary operation $\theta$ on $Map(A^{n},A)$ as follows: for any $f_{1},f_{2},...,f_{n},g\in Map(A^{n},A)$, we define $\theta(f_{1},f_{2},...,f_{n},g)$ as the composition of mappings $(f_{1},f_{2},...,f_{n}):A^{n}\rightarrow A^{n}$ and $g:A^{n}\rightarrow A$.

We have the mapping $F:A\rightarrow Map(A^{n},A)$ defined as $F(b)=\theta_{b}$, which sends an $n$-tuple $(a_{1},a_{2},...,a_{n})$ to $\theta(a_{1},a_{2},...,a_{n},b)$.

**   Lemma 3.1****.**

The following conditions are equivalent:

(a) The mapping $F$ preserves $\theta$;

(b) For any $a_{1},a_{2},...,a_{n},b_{1},b_{2},...,b_{n},c\in A$, one has

[TABLE]

[TABLE]

**   Definition 3.2****.**

Let $A$ be a set and let $\theta$ be an $(n+1)$-ary operation on $A$. $\theta$ is called 2-associative if the equivalent conditions of Lemma 3.1 are satisfied.

When $n=1$, the 2-associativity obviously is equivalent to the associativity in the usual sense.

*   Remark 3.3**.*

We left the term ”1-associativity” for the straightforward generalization of the usual associativity condition given by the move of parentheses, i.e. for the following condition: moving the internal symbol $\theta$ together with the attached parentheses in the expression $\theta(a_{1},a_{2},...,a_{n},\theta({b_{1},b_{2},...,b_{n},c}))$ to any place, one obtains the same element, for any $a_{1},a_{2},...,a_{n},b_{1},b_{2},...,b_{n},c\in A$.

An example of an 1-associative $(n+1)$-ary operation is given by the operation

[TABLE]

on a semigroup.

Let us now give examples of 2-associative operations.

**   Example 3.4****.**

(a) Let $A$ be a set. The operation $\theta$ on the set $Map(A^{n},A)$ defined above is 2-associative. More generally, let $\mathcal{C}$ be a category with finite products. Consider any object $C$ and $Mor(C^{n},C)=\{f:C^{n}\rightarrow C\}$, with the operation $\theta$ defined similarly to the case of a set $C$. The operation $\theta$ is 2-associative.

(b) Let $A$ be any set, and $i$ and $n$ be any natural numbers with $1\leq i\leq(n+1)$. Let

[TABLE]

Then $\theta$ is a 2-associative operation.

(c) Let A be a semigroup, and $i$ and $n$ be any natural numbers with $1\leq i\leq n$. Let us introduce the $(n+1)$-ary operation by

[TABLE]

Then $\theta$ is 2-associative.

(d) Let $A$ be the set of all $(n+1)\times(n+1)$ matrices. Let $\theta$ sends an $(n+1)$-tuple $(M_{1},M_{2},...,M_{n+1})$ to the matrix whose $i$th row is the $i$th row of the matrix $M_{i}$. This operation is 2-associative.

(e) Let $A$ be a commutative monoid such that the order of any element divides $(n-1)$. Then the operation $\theta$ defined by

[TABLE]

is both 1- and 2-associative.

**   Example 3.5****.**

Let $A$ be a set, and $\theta$ be a ternary operation. Let, for any $x\in A$, $\circ_{x}$ be the binary operation defined by

[TABLE]

Then $\theta$ is 2-associative if and only if

[TABLE]

for any $a,b,c,x,y\in A$. We have at least two such operations on a distributive lattice. One of them is defined by

[TABLE]

and the other by

[TABLE]

Recall that (3.3) is a promotodular operation in the algebraic theory of the variety of Boolean algebras **[2]**.

To construct further examples of 2-assocative protomodular algebras, below we give a simple lemma.

Let $\textbf{V}_{n}$ be the simplest protomodular variety, i.e. the variety with the signature $\mathfrak{F}_{n}$ containing only one $(n+1)$-ary operation symbol $\theta$, the binary operation symbols $\alpha_{1},\alpha_{2},...,\alpha_{n}$, and the constants $e_{1},e_{2},...,e_{n}$, where the identities are (2.1) and (2.2). Similarly, we denote by $\overline{\textbf{V}}_{n}$ the simplest semi-abelian variety.

**   Lemma 3.6****.**

Let $A$ be a nonempty set, and let $\theta:A^{n+1}\rightarrow A$ be any mapping.

(a) Let $e_{1},e_{2},...,e_{n}\in A$. There are binary operations $\alpha_{i}$ $(1\leq i\leq n)$ on $A$ such that

[TABLE]

is a $\textbf{V}_{n}$-algebra if and only if, for any $b\in A$, the mapping $\theta_{b}$ is surjective and

[TABLE]

In particular, if $\theta$ satisfies (3.5) and the identity

[TABLE]

then (3.4) is a $\textbf{V}_{n}$-algebra†††If $e_{1}=e_{2}=...=e_{n}$, then (3.5) and (3.6) imply that $A$ is one element set., for some $\alpha_{i}$ $(1\leq i\leq n)$.

(b) Let $e\in A$. There are binary operations $\alpha_{i}$ $(1\leq i\leq n)$ on $A$ such that

[TABLE]

is an $\overline{\textbf{V}}_{n}$-algebra if and only if for any $b\in A$, the mapping $\theta_{b}$ is surjective and

[TABLE]

**   Example 3.7****.**

Lemma 3.6 implies that if A from (c) of Example 3.4 is a group, then it can be turned into a 2-associative $\overline{\textbf{V}}_{n}$-algebra, for any $n$. Now consider any groups $A_{1},A_{2},...,A_{k}$. $A_{1,i_{1}}\times A_{2,i_{2}}\times...\times A_{k,i_{k}}$ can be turned into a 2-associative $\overline{\textbf{V}}_{n}$-algebra, for any naturals $i_{1},i_{2},...,i_{k}$ not exceeding $n$. Here $A_{j,i_{j}}$ denotes the set $A_{j}$ with the operation defined in (c) of Example 3.4 for $i=i_{j}$.

**   Example 3.8****.**

Any distributive lattice with top and bottom elements has the structure of a 2-associative $\textbf{V}_{2}$-algebra. Indeed, $\theta$ given by (3.3) satisfies (3.5) and (3.6).

**   Example 3.9****.**

Let $\mathcal{C}$ be a category with finite products and let $Mor_{\Delta}(C^{n},C)$ denote the set of retractions of the diagonal morphism $\triangle:C\rightarrow C^{n}$. It is closed under the operation $\theta$ considered in Example 3.4(a). Let $e_{i}$ be the $i$-th projection $\pi_{i}:C^{n}\rightarrow C$. Then (3.5) and (3.6) are satisfied since

[TABLE]

In this way $Mor_{\Delta}(C^{n},C)$ turns into a 2-associative $\textbf{V}_{n}$-algebra.

*   Remark 3.10**.*

One can show that the semi-abelian operations of Heyting algebras, Heyting semi-lattices, and locally Boolean distributive algebras given in [8], [1] are neither 1-associative nor 2-associative, provided that the algebras are not trivial. Moreover, neither of the operations given by (3.2) and (3.3) on a non-trivial lattice is 1-associative. However, as noted in Example 3.5, both operations are 2-associative.

*   Remark 3.11**.*

A Mal’cev operation can be 1-associative (for instance, in the variety of groups). However, it is 2-associative only for trivial algebras‡‡‡Indeed, we have $\mu(a,b,\mu(b,b,b))=a$ and $\mu(\mu(a,b,b),\mu(a,b,b),b)=b$..

To give further negative examples, let us introduce the notion of a strict protomodular algebra. First observe that the identity (2.2) implies that the equation

[TABLE]

has a solution, for any $a$, $b\in A$.

**   Lemma 3.12****.**

Let $A$ be an algebra from a protomodular variety. The following conditions are equivalent:

(i) $\theta_{b}$ is a bijection, for any $b\in A$;

(ii) the equation (3.9) has a unique solution for any $a,b\in A$;

(iii) the system of equations

[TABLE]

has a (unique) solution, for any $b,a_{i}\in A$ $(1\leq i\leq n)$.

(iv) the following identity is satisfied in $A$:

[TABLE]

for any $i$ (1$\leq i\leq n$).

**   Definition 3.13****.**

A protomodular operation $\theta$ is called strict if the equivalent conditions of Lemma 3.12 are satisfied. If it is clear which protomodular operation we are dealing with, then the term ”strict” will be referred to an algebra.

Obviously, if $n>2$, then any strict protomodular algebra is infinite.

**   Example 3.14****.**

$\textbf{V}_{n}$-algebras with the strict operation $\theta$ from the signature can be easily described. Consider any infinite set $A$, and, for any $b\in A$, choose a bijection $\theta_{b}:A\times A\times...\times A\rightarrow A$ such that

[TABLE]

Let us take

[TABLE]

where $\pi_{i}$ is the $i$-th projection $A\times A\times...\times A\rightarrow A$. In this way $A$ turns into a strict $\textbf{V}_{n}$-algebra. Obviously, any strict $\textbf{V}_{n}$-algebra can be given in this way.

Let V be a protomodular variety, and $\theta$ be its protomodular operation.

**   Example 3.15****.**

Let $n=1$. A protomodular operation $\theta$ on an algebra $A$ is strict if and only if the quadriple $\mathbb{A}=(A,\theta,\alpha_{1},1)$ is a left semiloop. This in particular implies that a 2-associative protomodular operation $\theta$ is strict if and only if $\mathbb{A}$ is a group.

*   Remark 3.16**.*

Let $n\geq 2$. Any strict protomodular algebra which is either 1-associative or 2-associative is trivial. Indeed, let $A$ be a strict 1-associative protomodular algebra. Consider any $a\in A$. From (2.3), we obtain

[TABLE]

Since $\theta_{a}$ is bijective, $a=e_{2}$, and hence the algebra $A$ is trivial.

Let now $A$ be a strict 2-associative algebra, and let $a_{1},a_{2},...,a_{n},b\in A$. From Lemma 3.18 below we obtain

[TABLE]

for any $i$ $(1\leq i\leq n)$. Now take any $i$ and $j$ with $1\leq i<j\leq n$. (3.12) implies that

[TABLE]

Hence $a_{j}=e_{j}$, and $A$ is trivial.

*   Remark 3.17**.*

Remark 3.16 and Example 3.5 imply that the protomodular operation (3.3) on a Boolean algebra mentioned in Section 2 is strict if and only if the algebra is trivial. One can show that, although the semi-abelian operations on non-trivial Heyting algebras, Heyting semi-lattices, and locally Boolean distributive lattices given in [8] and [1] are not 2-associative, they are not strict.

From (2.3) we obtain

**   Lemma 3.18****.**

Let $A$ be a 2-associative algebra. For any $a_{1},a_{2},...,a_{n}$ and $b\in A$, one has

[TABLE]

[TABLE]

4. Associative Semi-Abelian Algebras and Their Groups

From the fact that any set equipped with an associative binary operation, which has a left identity and left inverses, is a group (see, for instance, [5]), it immediately follows that an associative semi-abelian operation with $n=1$ is a group operation. In this section we study the question when an algebraic theory of a protomodular variety contains a group operation in the case of an arbitrary $n$.

From now on, unless specified otherwise, we will deal with an arbitrary protomodular variety.

**   Lemma 4.1****.**

Let $A$ be a 2-associative protomodular algebra. Then, for any $b,c\in A$, there is $a\in A$ such that

[TABLE]

The element $a$ can be taken as

[TABLE]

Proof.

The statement of this lemma immediately follows from the identities (3.1) and (2.2). Below we give another proof of the existence, it enables us to avoid cumbersome formulae.

Since the mapping $F:A\rightarrow Map(A^{n},A)$, $b\mapsto\theta_{b}$, preserves $\theta$, $F(A)$ is closed in $Map(A^{n},A)$ under the operation introduced in Section 3. Therefore, for any $b\in A$ , the mapping $f=\theta_{b}(\theta_{b},\theta_{b},...,\theta_{b}):A^{n}\rightarrow A$ lies in $F(A)$, and hence is equal to $\theta_{d}$ for some $d\in A$. This implies that $f$ is surjective. Then, for any $c\in A$, there is $(a_{1},a_{2},...,a_{n})\in A^{n}$ such that

[TABLE]

This implies (4.1) for

[TABLE]

∎

One can easily verify

**   Lemma 4.2****.**

Let $A$ be a 2-associative protomodular algebra. Then the binary operation

[TABLE]

is associative.

**   Proposition 4.3****.**

Let $\theta$ be a 2-associative semi-abelian operation on an algebra $A$. Then $A$ with the binary operation defined by (4.3) is a group. For any $b\in A$, we have

[TABLE]

[TABLE]

Proof.

The equality (2.3) implies that $e$ is a left unit. Now it suffices to apply Lemma 4.1, Lemma 4.2 and the above-mentioned statement from [5]. ∎

*   Remark 4.4**.*

Note that, for any 2-associative protomodular operation $\theta$, the operation given by (4.3) does not in general define the group structure. For the counter-example take the protomodular operation given by (3.3) on a Boolean algebra.

Proposition 4.3 immediately gives rise to

**   Corollary 4.5****.**

In the conditions of Proposition 4.3, we have:

(a) $\theta(a,a,...,a,e)=a$, for any $a\in A$;

(b) for any $b,c\in A$, there is a unique $a\in A$ with $\theta(a,a,...,a,b)=c$;

(c) for any $a,c\in A$, there is a unique $b\in A$ with $\theta(a,a,...,a,b)=c.$

**   Theorem 4.6****.**

For a variety V of universal algebras, the following conditions are equivalent:

(i) An algebraic theory of V contains a group operation;

(ii) An algebraic theory of V contains a constant and a Mal’cev operation $\mu$ which is associative in the sense of [7], i.e. satisfies the following identity

[TABLE]

(iii) An algebraic theory of V contains a semi-abelian operation which is 2-associative;

(iv) An algebraic theory of V contains a protomodular operation $\theta$ which satisfies the following identity

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for the corresponding binary operations $\alpha_{i}$.

Proof.

The equivalence of (i) and (ii) was given in [7]; we already have mentioned it in the Introduction. The equivalence of (i) and (iii) immediately follows from Proposition 4.3. For (i)$\Leftrightarrow$(iv) let us observe that a protomodular variety has the Mal’cev term $\mu(a,b,c)=\theta(\alpha_{1}(a,b),\alpha_{2}(a,b),...,\alpha_{n}(a,b),c)$. ∎

According to Proposition 4.3 we have the functor

[TABLE]

where V is a semi-abelian variety, $2$-$Ass\textbf{V}$ denotes the category of 2-associative V-algebras, while Grp denotes the category of groups; $R$ sends a 2-associative V-algebra to itself with the group structure introduced above. The functor $R$ obviously has a left adjoint.

At the end of the paper we give the description of 2-associative $\overline{\textbf{V}}_{n}$-algebras as groups with some additional structure. Let $n$ be a natural number with $n\geq 2$. We define an $n$-enriched group as a triple $(G,\gamma,(\alpha_{i})_{1\leq i\leq n})$, where $G$ is a group, $\gamma$ is a mapping (not necessarily a homomorphism) $G^{n}\rightarrow G$, and $\alpha_{i}$ is a binary operation on $G$, such that

[TABLE]

[TABLE]

and the following distributivity condition is satisfied:

[TABLE]

[TABLE]

Let $\textbf{EnrGrp}_{n}$ be the category, whose objects are $n$-enriched groups and morphisms are group homomorphisms preserving $\gamma$ and all $\alpha_{i}$.

Lemma 3.18 implies

**   Theorem 4.7****.**

The categories $2$-$Ass\overline{\textbf{V}}_{n}$ and $\textbf{EnrGrp}_{n}$ are isomorphic.

Proof.

Let $F:2$-$Ass\overline{\textbf{V}}_{n}\rightarrow\textbf{EnrGrp}_{n}$ be the functor sending an algebra $A$ to the set $A$ with the group structure described above. Moreover, let

[TABLE]

Then, as it follows from Corollary 4.5(a), $(G,\gamma,(\alpha_{i})_{1\leq i\leq n})$ is an enriched group. Consider the functor

[TABLE]

sending $(G,\gamma,(\alpha_{i})_{1\leq i\leq n})$ to the set $G$ equipped with the $(n+1)$-ary operation defined as

[TABLE]

Then $G$ turns into a 2-associative $\overline{\textbf{V}}_{n}$-algebra. One can easily verify that $F$ and $G$ are the mutually inverse functors. ∎

5. Acknowledgment

This work is supported by Shota Rustaveli National Science Foundation (Ref.: FR-18-10849).