Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces

1 Introduction

The quantum Yang–Baxter equation appeared in the work of Yang [49] and Baxter [2]. It is one of the basic equations in mathematical physics, and it laid the foundations of the theory of quantum groups [33]. Solutions of the Yang–Baxter equation are instrumental in the construction of semisimple Hopf algebras [23, 43] and provide examples of coloring invariants in knot theory [41]. More recently, the Yang–Baxter solution popped up in the theory of quantum computation [34, 50], where solutions of the Yang–Baxter equation provide so-called universal gates. One of the central open problems is to find all solutions of the Yang–Baxter equation. Let V be a vector space over a field K. Then a solution of the Yang–Baxter equation is a linear map R:V⊗V→V⊗V for which the following holds on V⊗3:

The simplest solutions are the solutions R induced by a linear extension of a mapping r:X×X→X×X, where X is a basis for V, satisfying the set-theoretic version of the Yang–Baxter equation, i.e., satisfying the following on X3:

In this case, r is said to be a set-theoretic solution of the Yang–Baxter equation (briefly, a solution). Drinfel’d, in [22], posed the question of finding these set-theoretic solutions. Denote for x,y∈X the element r⁢(x,y)=(λx⁢(y),ρy⁢(x)). One says that a set-theoretic solution r is left non-degenerate if λx is bijective for every x∈S, right non-degenerate if ρy is bijective for every y∈S, and non-degenerate if r is both left and right non-degenerate. If a solution is neither left nor right non-degenerate, then it is called degenerate. The first papers on set-theoretic solutions are those of Etingof, Schedler and Soloviev [24] and Gateva-Ivanova and Van den Bergh [27]. Both papers considered involutive solutions, i.e., solutions r where r2=id. Rump [44] introduced a new algebraic structure, braces, that generalizes radical rings and provides an algebraic framework. We provide the equivalent definition formulated by Cedó, Jespers and Okniński [12]. A triple (B,+,∘) is called a left brace if (B,+) is an abelian group and (B,∘) is a group such that for any a,b,c∈B it holds that

This new structure showed connections between the Yang–Baxter equation and ring theory, flat manifolds, orderability of groups, Garside theory, and regular subgroups of the affine group; see, for instance, [5, 6, 14, 15, 21, 25, 46]. Lu, Yan, and Zhu [38] and Soloviev [47] started the study of non-degenerate bijective solutions, not necessarily involutive. Almost all of the ideas used in the theory of involutive solutions can be transported to non-involutive solutions. We also mention Gateva-Ivanova and Majid [26] who studied and characterized most general set-theoretic solutions (braided sets) (X,r) in terms of the induced left and right actions of X on itself, and in terms of abstract matched pair properties of the associated braided monoid S⁢(X,r). The algebraic framework now is provided by skew left braces [28]. Let (B,+) and (B,∘) be groups on the same set B. If, for any a,b,c∈B, condition ($\diamond$) holds, the triple (B,+,∘) is called a skew left brace. Skew left braces and some of their applications are intensively studied; see, for instance, [8, 13, 20, 35, 45].

In [37], Lebed drew the attention on idempotent solutions. Indeed, using idempotent solutions and graphical calculus from knot theory, she provides a unifying tool to deal with several diverse algebraic structures such as free monoids, free commutative monoids, factorizable monoids, plactic monoids and distributive lattice. Examples and classifications of these solutions have been provided by Matsumoto and Shimizu [39] and by Stanovskỳ and Vojtěchovskỳ [48]. Moreover, Cvetko-Vah and Verwimp [19] provided cubic solutions with skew lattices. A cubic solution r is a solution such that r3=r; hence, this class includes both involutive and idempotent solutions. More generally, a more systematic approach to the study of solutions with finite order can be found in the recent [9, 10, 11]. Catino, Colazzo, and Stefanelli [7] and Jespers and Van Antwerpen [32] introduced the algebraic structure called left semi-brace to deal with solutions that are not necessarily non-degenerate or that are idempotent. Let (B,+) be a semigroup and let (B,∘) be a group. Then (B,+,∘) is called a left semi-brace if, for any a,b,c∈B, it holds that

where a- denotes the inverse a in (B,∘). If (B,+) is a left cancellative semigroup, then we call (B,+,∘) a left cancellative left semi-brace. This was the original definition by Catino, Colazzo and Stefanelli [7]. It has been shown that left semi-braces, under some mild assumption, provide set-theoretic solutions of the Yang–Baxter equation. Moreover, the associated solution is left non-degenerate if and only if the left semi-brace is left cancellative.

Out of algebraic interest, Brzeziński introduced left trusses [4] and left semi-trusses [3]. A quadruple (B,+,∘,λ) is called a left semi-truss if both (B,+) and (B,∘) are semigroups and λ:B×B→B is a map such that a∘(b+c)=(a∘b)+λ⁢(a,c). Clearly, the class of left semi-trusses contains all left semi-braces, rings, associative algebras and distributive lattices. This entails that it will prove difficult to present deep results on this class. However, one may examine large subclasses. In particular, Brzeziński [4] focused on left semi-trusses with a left cancellative semigroup (B,+) and a group (B,∘), and showed that such a left semi-truss is equivalent with a left cancellative semi-brace, and thus providing set-theoretic solutions of the Yang–Baxter equation, albeit known ones. In [40], Miccoli introduced almost left semi-braces, a particular instance of left semi-trusses, and constructed set-theoretic solutions associated with this algebraic structure. In [18], Colazzo and Van Antwerpen continued this study focusing on the subclass of brace-like left semi-trusses, i.e., left semi-trusses in which the multiplicative semigroup is a group and which includes almost left semi-braces. Concerning solutions, they showed that the solution one can associate with an almost left semi-brace is already the associated solution of a left semi-brace. In particular, this shows that brace-like left semi-trusses will not yield a universal algebraic structure that produces set-theoretic solutions.

In this paper, we focus on a new algebraic structure that includes left semi-braces and is an instance of left semi-trusses, which is on a different path with respect to brace-like left semi-trusses, called generalized left semi-brace. A triple (S,+,∘) is called a generalized left semi-brace if (S,+) is a semigroup, (S,∘) is a completely regular semigroup (or union of groups), and such that, for any a,b,c∈S, it holds

where a- denotes the (group) inverse of a in (B,∘). We prove that, under some mild assumptions, generalized left semi-braces provide solutions. In particular, elementary examples of generalized left semi-braces produce cubic solutions that cannot be obtained by skew lattices and left semi-braces. Also, we introduce a construction technique that provides generalized left semi-braces called the strong semilattice of generalized left semi-braces. This technique is inspired by the description of semigroups which are unions of groups due to Clifford [16].

Furthermore, we introduce a construction technique for solutions called the strong semilattice of solutions. This technique takes a family of disjoint sets {Xα∣α∈Y} indexed by a semilattice Y and solutions defined on these sets. Then, under some assumptions of compatibility, it allows one to construct a solution on the union of the sets Xα. We prove that the solutions provided by the strong semilattice of left semi-braces are a particular instance of a strong semilattice of solutions.

Finally, we prove that the strong semilattice of solutions is a useful tool to provide solutions of finite order. Indeed, the strong semilattice of solutions of finite order is a solution of finite order. Moreover, a solution r is of finite order if there exist a non-negative integer i and a positive integer p such that rp+i=ri, and the minimal integers that satisfy such relation are said to be index and period, respectively. We show that it is possible to determine the index and the period of the semilattice of solutions {rα∣α∈Y} as a function of the indexes and periods of rα. As a corollary of this result, we prove that solutions associated with strong semilattices of left semi-braces are not bijective, so they are clearly different from solutions obtained by left semi-braces.

2 Basic tools on left semi-braces

Let us briefly present some basic background information regarding left semi-braces. Most of the content of this section appears in [32]. In particular, we provide a different proof of [32, Corollary2.9] based on a result in semigroup theory due to Hickey [29] that gives a clear description of completely regular semigroups with middle units. Moreover, we add further information on the behavior of middle units of the additive semigroup of a left semi-brace. Finally, we present concrete examples of left semi-braces.

Let us start by recalling the definition of left semi-braces.

Throughout, 0 denotes the identity of the group (B,∘). Moreover, we call (B,+) and (B,∘) the additive semigroup and the multiplicative group of the left semi-brace (B,+,∘), respectively. Furthermore, if the semigroup (B,+) has a pre-fix, pertaining to some property of the semigroup, we will also use this pre-fix with the left semi-brace. Hence, the left semi-braces introduced in [7], where one works under the restriction that the semigroup (B,+) is left cancellative, will be called left cancellative left semi-braces.

Now, we recall that an element u of an arbitrary semigroup (S,+) is a middle unit of S if a+u+b=a+b for all a,b∈S. Thus, u+u is idempotent, but u itself need not be idempotent (see [17, p. 98]). This is not the case for the element 0 in the additive semigroup of a left semi-brace: the following proposition shows that 0 is an idempotent middle unit.

To show the following theorem, let us recall that an arbitrary semigroup (S,+) is said to be a rectangular group if it is isomorphic to the direct product of a group and a rectangular band. For more background and details on this topic, we refer the reader to [17].

A special case in which the additive semigroup is completely simple is when 0+B is a subgroup of (B,∘) (see [32, Theorem 2.8]). For instance, this is the case when B is finite.

The set of idempotents of a semigroup S will be denoted by E⁢(S). As a consequence of Theorem 2.3, we have that the additive semigroup (B,+) can be written as

that is, the direct sum of the left zero semigroup I:=E⁢(B+0), the group G:=0+B+0, and the right zero semigroup Λ:=E⁢(0+B). Moreover, the set of idempotents is E⁢(B)=I+Λ and it is a rectangular band.

For the sake of completeness, let us introduce a further property of middle units of left semi-braces that holds without restrictions on the additive semigroup. At first, we recall that the additive semigroup of a left semi-brace B does not contain a zero element if B has at least two elements (see [32, Lemma 2.3]).

In the following, we prove that middle units of the additive structure of an arbitrary left semi-brace are idempotents.

Following Ault’s paper [1, Theorem 1.8], by Proposition 2.4, we have that the additive semigroup of any left semi-brace contains a subsemigroup MB, called the semigroup of middle units of B, that is explicitly given by

where the inverse x′ of x means that x′=x′+x+x′ and x=x+x′+x.

Then MB is a subsemigroup of (B,+) that is a rectangular group. This is an interesting substructure of a left semi-brace, which is beyond the purpose of this paper and shall be studied elsewhere.

Now, having as reference [7, Example 2], we provide the following class of examples of completely simple left semi-braces that allow one to obtain solutions.

Let us examine special cases of the previous class of examples. Firstly, observe that if f≠id and g is not the constant map of value 0, then the semigroup (B,+) is neither left nor right cancellative. Moreover, note the following three cases:

Case 1. If g is the constant map of value 0, then

for all a,b∈B, i.e., B coincides with the left cancellative left semi-brace provided in [7, Example 2]. Moreover, the solution associated to B is given by

Case 2. If f=id, then

for all a,b∈B. In this case, the semigroup (B,+) is right cancellative. Indeed, if a+b=c+b, it follows that b∘g⁢(b-)∘a=b∘g⁢(b-)∘c, and so a=c. Moreover, it is easy to check that B is both a right and left semi-brace. In addition, note that the solution associated to B is given by

Case 3. If f=g, then

for all a,b∈B. Note that if a∈B, it holds

i.e., every element is idempotent with respect to the sum. In this case, the semigroup (B,+) is a rectangular band where ker⁡f=0+B and im⁡f=B+0. Moreover, the solution r associated to B, given by

is an idempotent solution, consistently with [32, Theorem 5.1] in the case in which G={0}.

The following is a class of examples of completely simple left semi-braces that, under suitable assumptions, give rise to solutions.

3 Definitions and examples

Braces, skew braces, and semi-braces were introduced to study set-theoretic solutions of the Yang–Baxter equation. The following definition generalizes these structure to the case in which the multiplicative structure is no more a group.

At first, we recall that a semigroup (S,∘) is completely regular if for any element a of S there exists a (unique) element a- of S such that

Conditions (3.1) imply that a0:=a∘a-=a-∘a is an idempotent element of (S,∘).

Let us note that if S is a generalized left semi-brace and a∈S, then the map

is an endomorphism of the semigroup (S,+) and λa∘b⁢(x)=(a∘b)0+λa⁢λb⁢(x) for all a,b,x∈S. Indeed, if a,b,x,y∈S, we have that

and

Of course, left semi-braces [7, 32] are examples of generalized left semi-braces. Moreover, a generalized left semi-brace can be obtained from every completely regular semigroup.

Unlike left semi-braces, a generalized left semi-brace S can have a zero element even if S has more than one element. Examples of such generalized left semi-braces can be easily obtained by any Clifford semigroup.

More generally, the previous generalized left semi-braces can be obtained through the following construction.

4 Solutions related to generalized left semi-braces

This section is devoted to provide a sufficient condition to obtain solutions through a generalized left semi-brace. To this end, we recall that if S is a left cancellative left semi-brace, then the map r:S×S→S×S given by

for all a,b∈S is a solution. Moreover, [32, Theorem 5.1] gives a sufficient condition to obtain that the map in (4.1) is still a solution for a left semi-brace, not necessarily left cancellative. In addition, in [10, Theorem 3], we state a necessary and sufficient condition to ensure that r is a solution.

Specifically, if (S,+,∘) is a left semi-brace, then the map r:S×S→S×S, defined by

is a solution if and only if

holds for all a,b,c∈S.

Let us remark that if S is a generalized left semi-brace with (S,+) being a right zero semigroup, then the map r as in (4.1) is a solution if and only if (a∘b)0=(a0∘b)0 holds for all a,b∈S. Observe that semigroups (S,∘) satisfying such a condition lie in the wide class of right cryptogroups; see [42]. In this way, we get new idempotent solutions that are of the form r⁢(a,b)=(a0,a∘b), different from those obtained in [37, 39, 19, 48].

Moreover, note that if (S,+,∘) is the generalized left semi-brace of Example 3.3, we obtain that

is a solution. In particular, if (S,∘) is commutative, clearly r⁢(a,b)=(a0∘b,b0∘a) and it is easy to verify that r is a cubic solution, i.e., r3=r.

Our aim is to show that if S=[Y;Sα,ϕα,β] is a strong semilattice of generalized left semi-braces such that every Sα satisfies condition (4.2), then the map in (4.1) is a solution. This result is a consequence of a more general construction technique on solutions we introduce in the following theorem.

The proof of Theorem 4.1 is technical, and for the sake of clarity, we present it in the next section. Now, as a consequence of this theorem, we obtain the following result.

5 Strong semilattices of set-theoretical solutions

This section aims to provide a proof of Theorem 4.1 and to give some examples of strong semilattices of solutions. Furthermore, we analyze strong semilattices of solutions with finite order.