This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.
The quantum Yang–Baxter equation appeared in the work of Yang  and Baxter . It is one of the basic equations in mathematical physics, and it laid the foundations of the theory of quantum groups . 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 . 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 for which the following holds on :
The simplest solutions are the solutions R induced by a linear extension of a mapping , where X is a basis for V, satisfying the set-theoretic version of the Yang–Baxter equation, i.e., satisfying the following on :
In this case, r is said to be a set-theoretic solution of the Yang–Baxter equation (briefly, a solution). Drinfel’d, in , posed the question of finding these set-theoretic solutions. Denote for the element . One says that a set-theoretic solution r is left non-degenerate if is bijective for every , right non-degenerate if is bijective for every , 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  and Gateva-Ivanova and Van den Bergh . Both papers considered involutive solutions, i.e., solutions r where . Rump  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 . A triple is called a left brace if is an abelian group and is a group such that for any 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  and Soloviev  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  who studied and characterized most general set-theoretic solutions (braided sets) 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 . The algebraic framework now is provided by skew left braces . Let and be groups on the same set B. If, for any , condition ($\diamond$) holds, the triple 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 , 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  and by Stanovskỳ and Vojtěchovskỳ . Moreover, Cvetko-Vah and Verwimp  provided cubic solutions with skew lattices. A cubic solution r is a solution such that ; 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  and Jespers and Van Antwerpen  introduced the algebraic structure called left semi-brace to deal with solutions that are not necessarily non-degenerate or that are idempotent. Let be a semigroup and let be a group. Then is called a left semi-brace if, for any , it holds that
where denotes the inverse a in . If is a left cancellative semigroup, then we call a left cancellative left semi-brace. This was the original definition by Catino, Colazzo and Stefanelli . 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  and left semi-trusses . A quadruple is called a left semi-truss if both and are semigroups and is a map such that . 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  focused on left semi-trusses with a left cancellative semigroup and a group , 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 , Miccoli introduced almost left semi-braces, a particular instance of left semi-trusses, and constructed set-theoretic solutions associated with this algebraic structure. In , 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 is called a generalized left semi-brace if is a semigroup, is a completely regular semigroup (or union of groups), and such that, for any , it holds
where denotes the (group) inverse of a in . 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 .
Furthermore, we introduce a construction technique for solutions called the strong semilattice of solutions. This technique takes a family of disjoint sets 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 . 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 , 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 as a function of the indexes and periods of . 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 . In particular, we provide a different proof of [32, Corollary2.9] based on a result in semigroup theory due to Hickey  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.
Let B be a set with two operations and such that is a semigroup and is a group. Then is said to be a left semi-brace if
for all , where is the inverse of a in .
Throughout, 0 denotes the identity of the group . Moreover, we call and the additive semigroup and the multiplicative group of the left semi-brace , respectively. Furthermore, if the semigroup 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 , where one works under the restriction that the semigroup is left cancellative, will be called left cancellative left semi-braces.
Now, we recall that an element u of an arbitrary semigroup is a middle unit of S if for all . Thus, 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.
Let B be a left semi-brace. Then the following assertions hold:
0 is a middle unit of .
0 is an idempotent of .
is a subgroup of .
is a subsemigroup of .
(i) See [32, Lemma 2.4 (1)].
(ii) Since, by (i), , we have that
Thus, since 0 is a middle unit, we obtain
(iii) If , by (ii) we have that
(iv) By (ii), it is clear that is not empty. Moreover, if then, using (i), we get the equalities
(v) See [32, Lemma 2.6 (iii)]. ∎
To show the following theorem, let us recall that an arbitrary semigroup 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 .
Let B be a completely simple left semi-brace. Then the additive semigroup of B is a rectangular group.
The thesis follows by [29, Corollary 3.5], which states that any completely simple semigroup with a middle unit is a rectangular group. ∎
A special case in which the additive semigroup is completely simple is when is a subgroup of (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 . As a consequence of Theorem 2.3, we have that the additive semigroup can be written as
that is, the direct sum of the left zero semigroup , the group , and the right zero semigroup . Moreover, the set of idempotents is 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.
Let B be a left semi-brace. Then every middle unit is an idempotent of the semigroup .
Since 0 is idempotent, we have that
Since e is a middle unit, it follows that
which is our assertion. ∎
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 , called the semigroup of middle units of B, that is explicitly given by
where the inverse of x means that and .
Then is a subsemigroup of 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 be a group with identity 0, and let be idempotent endomorphisms of such that . Let us consider the following operation:
for all . It is easy to check that is a completely simple left semi-brace. Now, observe that the map ρ is an anti-homomorphism from the group into the monoid , where denotes the monoid of the functions from B into itself. Indeed, is given by
for all . Thus, by [32, Proposition 2.14], the semigroup is completely simple.
In addition, since ρ is an anti-homomorphism of the group , we obtain that the map defined by in [32, Theorem 5.1] is a solution. Note that . Therefore, r is explicitly given by
for all .
Let us examine special cases of the previous class of examples. Firstly, observe that if and g is not the constant map of value 0, then the semigroup 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 , 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 , then
for all . In this case, the semigroup is right cancellative. Indeed, if , it follows that , and so . 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 , then
for all . Note that if , it holds
i.e., every element is idempotent with respect to the sum. In this case, the semigroup is a rectangular band where and . Moreover, the solution r associated to B, given by
is an idempotent solution, consistently with [32, Theorem 5.1] in the case in which .
The following is a class of examples of completely simple left semi-braces that, under suitable assumptions, give rise to solutions.
Let G, H be two groups, let and consider the group where
for all , i.e., the classical Zappa–Szép product of G and H (see ) that has identity . Let φ be a map from G into H such that and define the following operation on B:
for all . It is easy to check that the structure is a left semi-brace. Let us note that is a left group. Moreover, in is idempotent with respect to the sum if and only if . In addition, if , we have that
Hence is a sub-semigroup of and it is also a left zero semigroup. Note also that
i.e., is a right identity with respect to the sum. Furthermore, we have that
for all . One can check that ρ is an anti-homomorphism if and only if it holds
for all and . Moreover, by the characterization [10, Theorem 3], one can verify that the map r in [32, Theorem 5.1] associated to the left semi-brace B is a solution if and only if
holds for all and . On the other hand, note that, if and , considering in (2.2), we obtain
Hence (2.1) is satisfied.
Now, let G be the cyclic group of 2 elements, let H be the cyclic group of 3 elements, and let φ be the constant map of value 1 from G into H. Hence, if is the cyclic group , then condition (2.1) trivially holds, and hence r is a solution. Instead, if is the symmetric group , then (2.1) is not satisfied; equivalently, (2.2) does not hold, and hence r is not a solution.
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 is completely regular if for any element a of S there exists a (unique) element of S such that
Conditions (3.1) imply that is an idempotent element of .
Let S be a set with two operations and such that is a semigroup (not necessarily commutative) and is a completely regular semigroup. Then we say that is a generalized left semi-brace if
for all . We call and the additive semigroup and the multiplicative semigroup of S, respectively.
A generalized right semi-brace is defined similarly, replacing condition (3.2) by
for all .
A generalized two-sided semi-brace is a generalized left semi-brace that is also a generalized right semi-brace with respect to the same pair of operations.
Let us note that if S is a generalized left semi-brace and , then the map
is an endomorphism of the semigroup and for all . Indeed, if , we have that
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.
If is an arbitrary completely regular semigroup and is a right zero semigroup (or a left zero semigroup), then is a generalized two-sided semi-brace.
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.
If is a Clifford semigroup, which is a completely regular semigroup where all idempotent elements are central, then , where for all , is a generalized two-sided semi-brace.
More generally, the previous generalized left semi-braces can be obtained through the following construction.
Let Y be a (lower) semilattice, let be a family of disjoint generalized left semi-braces. For each pair of elements of Y such that , let be a homomorphism of generalized left semi-braces such that the following conditions hold:
is the identical automorphism of . for every .
for all such that .
Then endowed by the addition and the multiplication defined by
for any and , is a generalized left semi-brace. Such a generalized left semi-brace is said to be the strong semilattice Y of the generalized left semi-brace and is denoted by .
First note that is a semigroup and is a completely regular semigroup. Now, let , , and . Set , , , and . It follows that
where the last equality holds since is a generalized left semi-brace. Moreover,
Therefore, S is a generalized left semi-brace. ∎
If is a strong semilattice Y of left semi-braces , then is a strong semilattice of groups, and hence, by [30, Theorem 4.2.1], is a Clifford semigroup.
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 given by
for all 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 is a left semi-brace, then the map , defined by
is a solution if and only if
holds for all .
Let us remark that if S is a generalized left semi-brace with being a right zero semigroup, then the map r as in (4.1) is a solution if and only if holds for all . Observe that semigroups satisfying such a condition lie in the wide class of right cryptogroups; see . In this way, we get new idempotent solutions that are of the form , different from those obtained in [37, 39, 19, 48].
Moreover, note that if is the generalized left semi-brace of Example 3.3, we obtain that
is a solution. In particular, if is commutative, clearly and it is easy to verify that r is a cubic solution, i.e., .
Our aim is to show that if is a strong semilattice of generalized left semi-braces such that every 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.
Let Y be a (lower) semilattice, let be a family of disjoint solutions indexed by Y such that for each pair with there is a map . Let X be the union
and let be the map defined by
for all and . Then is a solution if the following conditions are satisfied:
is the identity map of for every .
for all such that .
for all such that .
We call the pair a strong semilattice of solutions indexed by Y.
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.
Let be a strong semilattice of generalized left semi-braces. Then, if satisfies (4.2) for every , then the map defined by
for all is a solution.
For any , let be the solution associated to the left semi-brace , i.e., the map defined by . Since S is a strong semilattice of left semi-braces, by Proposition 3.4, is the identical automorphism of and for all such that . Hence, conditions (i) and (ii) in Theorem 4.1 are satisfied. Moreover, let such that . Since, is a homomorphism of left semi-braces, for all it follows that
Hence Theorem 4.1 (iii) holds. Therefore, according to Theorem 4.1, we shall consider the strong semilattice Y of solutions , i.e., the map r defined by
for all , . Finally, note that, by Proposition 3.4,
for all , . ∎
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.
Proof of Theorem 4.1.
At first, note that if and are the maps from into itself that define every solution , i.e., is written as
for all , then condition (iii) is equivalent to the following equalities:
for all such that and . In addition, let us observe that if and , then the two components of the map r, i.e.,
lie in , consistently with the second part of the subscript of the maps ϕ. To avoid overloading the notation, hereinafter we will write the previous elements as
Now, we verify that r is a solution proving that the relations
are satisfied for all . For this purpose, let be elements of , , , respectively, and assume and
we have that
it follows that
Moreover, it holds
Hence we obtain that . Now, setting
it follows that
Furthermore, since U and V lie in , we have
As seen before, and . Thus
Consequently, since is a solution, we obtain that .
Finally, since W and Z lie in , note that
As seen before, . In addition, we have
It follows that
Moreover, it holds
and hence . Therefore, the map r is a solution. ∎
One can use strong semilattices of solutions with ’s of finite order to produce examples of new solutions with finite order.
(i) Let X be a semilattice of sets indexed by Y such that conditions (i) and (ii) of Theorem 4.1 are satisfied and let be the twist map on for every . Then, if , we have that
for all , i.e., condition (iii) of Theorem 4.1 holds. Hence, the strong semilattice of solutions is such that . Indeed, if and , assuming , we have that . Hence , and so
(ii) Let be a semilattice of sets such that , let c be a fixed element of , and let for every . Let be the twist map on and let be the idempotent solution on defined by for all . Then, if , we have that
Hence the assumptions of Theorem 4.1 are satisfied. Moreover, the strong semilattice of solutions is such that . Indeed, if and , since , it follows that
Therefore, we obtain that .
(iii) Let be a semilattice of sets such that , let c be a fixed element of , and let for every . Let f be an idempotent map from into itself, , and let be the map from into itself defined by for all . Thus, is a solution such that . Let be the idempotent solution defined by for all . Then, if , we obtain that
Thus the hypotheses of Theorem 4.1 are satisfied. Moreover, the strong semilattice of solutions is such that . Indeed, if and , since , it follows that
and clearly . Therefore, .
To investigate strong semilattices of solutions with finite order, we need the notions of the index and the period of a solution r that are
respectively. These definitions of the index and the order are slightly different from the classical ones (cf. [31, p. 10]), but they are functional to distinguish bijective solutions from non-bijective ones. For more details, we refer the reader to .
In the following theorem, we show that, given a semilattice Y of finite cardinality, the strong semilattice of solutions indexed by Y is of finite order if and only if solutions are. Furthermore, it allows for establishing the order of the strong semilattice of solutions if the index and the period of solutions are known. Conversely, the index and the period of a strong semilattice of solutions give us upper bounds of the indexes and periods of solutions .
Let be a strong semilattice of solutions indexed by a finite semilattice Y. Then is a solution with finite order on for every if and only if r is a solution with finite order. More precisely, the index of r is
and the period is
At first, suppose that is a solution with finite order for every . Let and . If and , setting , we have that
Consequently, if , i.e., is bijective for every , we obtain that
Therefore, and clearly the index of r is 1 by the assumption on i. Now, assume that and that for a certain . If , for a positive integer h, in particular, we have that . Since for a certain natural number q, it follows that
Hence , and so .
Now, we proceed to determine the period of r. If for a natural number m, then for every . Consequently, divides for every . Thus divides , i.e., divides . Therefore, , and hence . Conversely, suppose that the solution r is with finite order and set and . If , since , we obtain that . Therefore, is a solution with finite order. Clearly, we have that is less than i and divides p. ∎
Let us note that if is a strong semilattice of non-bijective solutions such that and for every , then r is still a solution of index i and period n, also in the case of an infinite semilattice Y. Indeed, one can prove this statement by similar computations used for the non-bijective case in the proof of Theorem 5.2.
Funding statement: This work was partially supported by the Dipartimento di Matematica e Fisica “Ennio De Giorgi” – Università del Salento. The second author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders), Grant G016117. The authors are members of GNSAGA (INdAM).
We would like to thank the referee for the accurate review.
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