Multisemigroups with multiplicities and complete ordered semi-rings

Motivated by appearance of multisemigroups in the study of additive $2$-categories, we define and investigate the notion of a multisemigroup with multiplicities. This notion seems to be better suitable for applications in higher representation theory.


Introduction s1
Abstract 2-representation theory originates from the papers [BFK, Kh, CR] and is nowadays formulated as study of 2-representations of additive k-linear 2-categories, where k is the base field, see e.g. [Ma] for details. Various aspects of general 2-representation theory of abstract additive k-linear 2-categories were studied in the series [MM1,MM2,MM3,MM4,MM5,MM6] of papers by Mazorchuk and Miemietz. An important role in this study is played by the so-called multisemigroup of an additive k-linear 2-category which was originally introduced in [MM2].
Recall that a multisemigroup is a set S endowed with a multioperation, that is a map * : S × S → 2 S which satisfies the following associativity axiom: for all a, b, c ∈ S (see [KM] for more details and examples). The original observation in [MM2] is that the set S of isomorphism classes of indecomposable 1-morphisms in an additive k-linear 2-category C has the natural structure of a multisemigroup, given as follows: for two indecomposable 1-morphisms F and G, we have where [F] stands for the isomorphism class of F and • denotes composition in C . We refer the reader to [MM2] for details. Combinatorics of this multisemigroup reflects and encodes various structural properties of the underlying additive k-linear 2category and controls major parts of the 2-representation theory of the latter, see [MM1,MM2,MM3,MM4,MM5,MM6] for details.
However, this notion of a multisemigroup of an additive k-linear 2-category has one disadvantage: it seems to forget too much information. In more details, it only records information about direct summands appearing in the composition F • G, however, it forgets information about multiplicities with which these direct summands appear. As as result, the multisemigroup of an additive k-linear 2category can not be directly applied to the study of the split Grothendieck category of C and linear representations of the latter.
It is quite clear how one can amend the situation: one has to define a weaker notion than a multisemigroup which should keep track of multiplicities in question. This naturally leads to the notion of multisemigroups with multiplicities, or multi-multisemigroups which is the object of the study in this paper (the idea of such object is mentioned in [MM2,Remark 8] without any details). Although the definition is rather obvious under natural finiteness assumption, the full generality setup has some catches and thus requires some work. The main aim of the present paper is to analyze this situation and to propose a "correct" definition of a multi-multisemigroup. The main value of the paper lies not in the difficulty of the presented results but rather in the thorough analysis of the situation which explores various connections of the theory we initiate. Our approach utilizes the algebraic theory of complete semirings.
The paper is organized as follows: in Section 2 we outline in more details the motivation for this study coming from the higher representation theory. In Section 3 we collect all notions and tool necessary to define our main object: multimultisemigroups, or, how we also call them, multisemigroups with multiplicities bounded by some cardinal. Section 4 ties back to the original motivation and is devoted to the analysis of multisemigroups with multiplicities appearing in the higher representation theory. In Section 5 we give some explicit examples. In Section 6 we discuss multi-multisemigroups for different sets of multiplicities and connection to twisted semigroup algebras. Finally, in Section 7, we describe multimultisemigroups as algebras over complete semirings.
A 2-category is a category enriched over the category of small categories. In other words, a 2-category C consists of • objects, • small categories of morphisms, • identity objects in the appropriate morphisms categories, • bifunctorial composition and all these data is supposed to satisfy all the obvious axioms. The canonical example of a 2-category is the category Cat of small categories where • objects are small categories, • morphisms are categories where objects are functors and morphisms are natural transformations of functors, • identities are the identity functors, • composition is composition of functors.
As usual, if C is a 2-category and i, j ∈ C , the objects in C(i, j) are called 1morphisms and morphisms in C (i, j) are called 2-morphisms. Composition of 2morphisms inside C(i, j) is called vertical, while composition of 2-morphisms in C is called horizontal.
Let k be a field. We will say that a 2-category C is k-admissible provided that, • for any i, j ∈ C , the category C(i, j) is k-linear, idempotent split and Krull-Schmidt, • composition is k-bilinear.
For example, let A be a finite-dimensional associative algebra and C a small category equivalent to A-mod, then the 2-full subcategory R (A,C) of Cat with unique object C and whose 1-morphisms are right exact endofunctors on C, is k-admissible. The reason for this is the fact that R (A,C) (C, C) is equivalent to the category of A-Abimodules, see [Ba] for details.
s2.2 2.2. Grothendieck category of a k-admissible 2-category. Let C be an additive category. Then the split Grothendieck group [C] ⊕ of C is defined as the quotient of the free abelian group generated by [X], where X ∈ C, modulo the relations If C is idempotent split and Krull-Schmidt, then [C] ⊕ is isomorphic to the free abelian group generated by isomorphism classes of indecomposable objects in C.
Let C be a k-admissible 2-category. The associated Grothendieck category [C ] ⊕ , also called the decategorification of C , is defined as the category such that • [C] ⊕ has the same objects as C , • identity morphisms in [C] ⊕ are classes of the corresponding identity 1morphisms in C , • composition in [C] ⊕ is induced from the composition in C.
We note that the category [C ] ⊕ is, by definition, preadditive, but not additive in general (as, in general, no coproduct of objects in C was assumed to exist).

ex1
Example 1. Let S be a finite semigroup with an admissible partial order ≤. Define the 2-category S as follows: • S S has one object i; • 1-morphisms in S S are elements from S; • composition of 1-morphisms is given by multiplication in S; • for two 1-morphisms s, t ∈ S, we have • vertical composition of 2-morphism is defined in the unique possible way which is justified by transitivity of <; • horizontal composition of 2-morphism is defined in the unique possible way, which is justified by admissibility of <.
For a field k, define the k-linearization S k of S as follows, see [GM2,Subsection 4.3] for details: • S k has one object i; • 1-morphisms in S k are formal finite direct sums of 1-morphisms in S; • 2-morphisms in S k are appropriate matrices whose entries are in kh s,t ; • compositions in S k are induced from those in S using k-bilinearity.
The 2-category S k is, by construction, k-admissible. Moreover, the decategorification [S k ] ⊕ of this 2-category • has one object i; • the endomorphism ring [S k ] ⊕ (i, i) of the object i is isomorphic to the integral semigroup ring Z[S].
For example, the category S k constructed in Example 1 is finitary (by construction and using the fact that S is finite). Here we see that the multioperation * is genuinely multi-valued.
s2.5 2.5. Multisemigroup and decategorification. Comparing (2) with (3), it is easy to see that the information encoded by the multisemigroup, that is (3), is not enough to recover the "associative algebra structure" which exists on the level of the Grothendieck decategorification presented in (2). The essential part of the information which got lost is the exact values of non-zero multiplicities with which indecomposable 1-morphism appear in composition of two given indecomposable 1-morphisms.
One can say that the situation is even worse. Let us try to use (3) to define some associative algebra structure on the abelian group Z[S 2 ]. The only reasonable guess would be to define, on generators, an operation ⋄ as follows: x ⋄ y = z∈x * y z and then extend this to Z[S 2 ] by bilinearity. However, this is not associative, for example, To have associativity, we should have considered B[S 2 ], where B is the Boolean semiring. This will be explained in details later.
Therefore, if we want to define some discrete object which we could use to recover the associative algebra structure given by the Grothendieck decategorification, we need to keep track of multiplicities. This naturally leads to the notion of multisemigroups with multiplicities.
3. Multisemigroups with multiplicities s3 s3.1 3.1. Semirings. A semiring is a weaker notion than that of a ring and the difference is that it is only required to form a commutative monoid (not a group) with respect to addition. More precisely, a unital semiring is a tuple (R, +, ·, 0, 1), where • R is a set; • + and · are binary operations on R; • 0 and 1 are two different elements of R.
These data is required to satisfy the following axioms.
• (R, +, 0) is a commutative monoid with identity element 0; • (R, ·, 1) is a monoid with identity element 1; • multiplication distributes over addition both from the left and from the right; We refer to [Go, Ka] for more details.
Here are some examples of semirings: • Any unital ring is a unital semiring.
• If R is a semiring, then the set Mat n×n (R) of n×n matrices with coefficients in R forms a semiring with respect to the usual addition and multiplication of matrices.
• For any nonempty set X, we have the semiring B X := (B X , ∪, ∩, ∅, X). This semiring is isomorphic to where B (x) = B, a copy of the Boolean semiring B indexed by x.
Semirings and homomorphisms form a category, denoted by SRing.
s3.2 3.2. Complete semirings. A commutative monoid (S, +, 0) is called complete provided that it is equipped, for any indexing set I, with the sum operation i∈I such that We refer the reader to [He] for more details.
A semiring (R, +, ·, 0, 1) is called complete provided that • (R, +, 0) is a complete monoid; • multiplication distributes over all operations i∈I on both sides, that is Given two complete semirings Complete semirings and homomorphisms form a subcategory in SRing, denoted by CSRing.
Here are some examples of complete semirings: • Any bounded complete join-semilattice is a complete commutative semiring.
• (B X , ∪, ∩, ∅, X), for some set X, where i∈I is the usual union.
• The set of open sets for a topological space X, with respect to union and intersection.
• Unital quantales with join as addition and the underlying associative operation as multiplication.
It is very tempting to add to the above the following "example": all cardinal numbers form a complete semiring with respect to the usual addition (disjoint union) and multiplication (Cartesian product) of cardinals. There is one problem with this "example", namely, the fact that cardinals do not form a set but, rather, a proper class. This problem can be overcome in an artificial and non-canonical way described in the next example. This examples is separated from the rest due to its importance in what follows.

ex3
Example 3. For a fixed cardinal κ, let Card κ denote the set of all cardinals which are not greater than κ. Then Card κ has the structure of a complete semiring where • addition (of any number of elements) is given by disjoint union with convention that all cardinals greater than κ are identified with κ; • multiplication is given by Cartesian product with convention that all cardinals greater than κ are identified with κ.
Note that the Boolean semiring B is isomorphic to Card 1 . • A is a set; • µ : A → Z ≥0 is a function, called the multiplicity function.
A natural, more general, notion is that of a genuine multiset, which is a pair (A, µ), where • A is a set; • µ is the multiplicity function from A to the class of all cardinals. s3.4 3.4. Multi-Booleans. Recall that, given a base set X, the Boolean B(X) = B X of X is the set of all subsets of X. This can be identified with the set of all functions from X to the Boolean semiring B. In this way, B(X) gets the natural structure of a complete semiring with respect to the union and intersection of subsets. The additive identity is the empty subset while the multiplicative identity is X. Note that we can also consider the dual Boolean of X which is the set of all functions from X to the dual Boolean semiring B * . This gets the natural structure of a complete semiring with respect to the intersection and union of subsets. The additive identity is X while the multiplicative identity is the empty subset.
The above point of view allows us to generalize the definition of the Boolean to multiset structures. Given a base set X, define the full multi-Boolean of X is the class of all functions from X to the class of all cardinal numbers. To create any sensible theory, we need sets. This motivates the following definition.
Given a base set X and a cardinal number κ ≥, define the κ-multi-Boolean B κ (X) of X is the set of all functions from X to the complete semiring Card κ . By construction, B κ (X) is equipped with the natural structure of a complete semiring. Also, we have B(X) = B 1 (X).
Clearly κ = 0 would give us a singelton, on which no semi-ring structure exists. From now on we agree that any cardinal κ in this paper is greater than or equal to 1.
Unfortunately, for κ > 1, the natural complete semiring structure on B κ (X) does not correspond to the usual set-theoretic notions of union and intersection of multisets. Indeed, the multiplicity analogue of the intersection of multisets is the arithmetic operation of taking minimum, while the multiplicity analogue of the union of multisets is the arithmetic operation of taking maximum. These differ from the usual addition and multiplication in Card κ , if κ > 1. • S is a non-empty set; • µ : S × S → B κ (S), written (s, t) → µ s,t : S → Card κ ; such that the following distributivity requirement is satisfied: for all r, s, t ∈ R, we have eq5 eq5 (4) i∈S µ s,t (i)µ r,i = j∈S µ r,s (j)µ j,t .
We note that here, for a cardinal λ and a function ν : S → Card κ , by λν we mean the function from S to Card κ defined as or, in other words, this is just adding up λ copies of ν.
The informal explanation for the requirement (4) is as follows: the left hand side corresponds to the "product" r * (s * t). Here s * t gives µ s,t , which counts every i ∈ S with multiplicity µ s,t (i). The result of r * (s * t), written when we distribute r * over all such i ∈ S and taking multiplicities into account, gives exactly the left hand side in (4). Similarly, the right hand side corresponds to the "product" (r * s) * t.
If κ is clear from the context, we will sometimes use the shorthand multi-multisemigroup instead of "multisemigroup with multiplicities bounded by κ".
Here are some easy examples of multisemigroup with multiplicities: • A usual multisemigroup is a multisemigroup with multiplicities bounded by one.
• For any κ and any λ < κ, the set {a} has the structure of a multisemigroup with multiplicities bounded by κ, if we set µ a,a = λ. Moreover, these exhaust all such structures on {a}.
Here is a more involved example: ex4 Example 4. Let A be a finite dimensional R-algebra with a fixed basis {a i : i ∈ I} such that a i · a j = s∈I µ s i,j a s and all µ s i,j ∈ Z ≥0 . Then (I, µ), where we define µ i,j (s) := µ s i,j , is a multisemigroup with multiplicities bounded by ω, the first infinite cardinal. This follows from the associativity of multiplication in A via the computation which is equivalent to (4) in this case since basis elements are linearly independent.
Let (S, µ) and (S ′ , µ ′ ) be two multisemigroups with multiplicities bounded by κ. We will say that they are isomorphic provided that there is a bijection ϕ : S → S ′ such that µ ′ = ϕ • µ.
Let (S, µ) be a multisemigroups with multiplicities bounded by κ. Let S ⋄ denote the set of all words in the alphabet S of length at least two. Define the map µ : S ⋄ → B κ (S) recursively as follows: (1) µ st = µ s,t , if s, t ∈ S; (2) if w = sx, where x has length at least two, then set eq5-1 eq5-1 The definition of µ does not really depend on our choice of prefix above (in contrast to suffix), as follows from the following statement.
prop17 Proposition 5. If w ∈ S ⋄ has the form w = xs, where x has length at least two, then eq5-2 eq5-2 Proof. Let w = s 1 s 2 . . . s k , where k ≥ 3. Then the recursive procedure in (5) results in The recursive procedure in (6) The expression (7) it transferred to (8) using a repetitive application of (4). The claim follows. 3.6. Finitary multisemigroups with multiplicities. We will say that a multisemigroup (S, µ) with multiplicities bounded by κ is finitary provided that • κ = ℵ 0 ; • µ r,s (t) = ℵ 0 for all r, s, t ∈ S; • |{t ∈ S : µ r,s (t) = 0}| < ℵ 0 for all r, s ∈ S. s3.7 3.7. Multi-multisemigroup of a k-admissible 2-category. Let C be a k-admissible 2-category. Consider the set S(C ) of isomorphism classes of indecomposable 1-morphisms in C . For F, G, H ∈ S(C ), define µ F,G (H) to be the multiplicity of H as a direct summand in the composition F • G.
Proof. We only have to check (4) in this case, as the rest follows by construction from k-admissibility of C . For F, G, H, K ∈ S(C ), the multiplicity of K in (F •G)•H is given by In turn, the multiplicity of K in F • (G • H) is given by which proves (4) in this case.

ex8
Example 7. For the 2-category S 3 in Example 2, the multi-multisemigroup structure on S(C ) is fully determined by (2). For instance, the function µ st,st has the following values: x : e s t st ts sts µ st,st (x) : 0 0 0 1 0 2. The function µ ts,sts has the following values: As usual, we identify subsets in a set X with B X . The following proposition says that the multi-multisemigroup of C has enough information to recover the multisemigroup of C .
prop11 Proposition 8. Let C be a k-admissible 2-category. Then, for any Proof. This follows directly from the definitions. (9) s · t := r∈S µ s,t (r)r.
Proof. We need to show that (r · s) · t = r · (s · t), for all r, s, t ∈ S. Using (9) and k-bilinearity of ·, this reduces exactly to the axiom (4).
s4.3 4.3. Grothendieck ring of a k-admissible 2-category. Let C be a small kadmissible 2-category. The Grothendieck ring Gr(C ) of C is defined as follows: • elements of Gr(C ) are elements in the free abelian group generated by isomorphism classes of indecomposable 1-morphisms; • addition in Gr(C ) is the obvious addition inside the free abelian group; • multiplication in Gr(C ) is induced from composition in C using biadditivity.
The ring Gr(C ) is unital if and only if C has finitely many objects. Otherwise it is locally unital, where local units correspond to (summands of) the identity 1-morphisms in C.
An alternative way to look at Gr(C ) is to understand it as the ring associated with the preadditive category [C ] ⊕ in the obvious way. Conversely, [C ] ⊕ is the variation of the ring Gr(C ) which has several objects, cf. [Mi]. Proposition 10. Let C be a finitary 2-category and k a field. Then there is a canonical isomorphism of k-algebras, Proof. We define the map ψ : k ⊗ Z Gr(C ) → k[S(C )] as the k-linear extension of the map which sends an isomorphism class of indecomposable 1-morphisms in C to itself. This map is, clearly, bijective. Moreover, it is a homomorphism of rings since, on both sides, the structure constants with respect to the k-basis, consisting of isomorphism class of indecomposable 1-morphisms in C, are given by non-negative integers µ F,G (H) as defined in Subsection 3.5. The claim of the proposition follows.
Altogether, for a finitary 2-category C , we have the following picture (S(C ), µ) , , where arrow show in which direction we can recover information.

5.
Some explicit examples of multi-multisemigroups of finitary 2-categories s5 s5.1 5.1. Projective functors for finite dimensional algebras. Let k be an algebraically closed field and A a connected, basic, non-semi-simple, finite dimensional, unital k-algebra. Let C be a small category equivalent to A-mod. Following [MM1,Subsection 7.3], we define the 2-category C A as a subcategory in Cat (not full) such that: • C A has one object i, which we identify with C; • 1-morphisms in C A are functors isomorphic to direct sums of the identity functors and functors of tensoring with projective A-bimodules; • 2-morphisms in C A are natural transformations of functors.
Note that all 1-morphisms in C A are, up to isomorphism, functors of tensoring with A-bimodules. For simplicity we will just use certain bimodules to denote the corresponding isomorphism classes of 1-morphisms.
The essential part of the composition in C A is given by as follows from the computation The above implies that S(C A ) = {½, F i,j : i, j = 1, 2, . . . , k} and the multiplicity function defining the multi-multisemigroup structure on S(C A ) is given by Note also that, in this case, the multioperation in the multisemigroup (S(C A ), * ) is, at most, single valued. By adding, if necessary, an external element 0, we can turn (S(C A ), * ) into a genuine semigroup. s5.2

Soergel bimodules for finite Coxeter groups.
Another prominent example of a finitary 2-category is the finitary 2-category of Soergel bimodules. Let W be a finite Coxeter group with a fixed geometric representation. To these data, one associates the so-called 2-category S W of Soergel bimodules over the coinvariant algebra of the geometric representation, see [So] and [MM1,Subsection 7.1]. This is a finitary 2-category. This 2-category categorifies the integral group ring of W in the sense that there is an isomorphism between the ring [C ] ⊕ (i, i) and the ring Z[W ] given in terms of the Kazhdan-Lusztig basis in Z[W ], see [KL]. Therefore the multi-multisemigroup (S(S W ), µ) records exactly the information about the structure constants of the ring Z[W ] with respect to the Kazhdan-Lusztig basis.
As far as we know, there is no explicit combinatorial formula for such structure constants, however, they can be determined using a recursive algorithm.
In the special case of a Dihedral group D n , where n ≥ 3, W = D n = {s, t : s 2 = t 2 = (st) n = e}, the Kazhdan-Lusztig basis has particularly simple form. Elements of the group D n can be listed as D n = {e, s, t, st, ts, . . . , w 0 }, where w 0 = stst · · · = tsts . . . , where the length of both words is n. Then, for each w ∈ D n , the corresponding Kazhdan-Lusztig basis element w ∈ Z[D n ] is the sum of w with all elements of strictly smaller length.
Let l : D n → Z ≥0 be the length function with respect to generators s and t. A direct calculation then shows that l(sw) < l(w); sw, w = e or w = t; sw + tw, otherwise; l(tw) < l(w); tw, w = e or w = s; sw + tw, otherwise.
This already shows that the multi-multisemigroup structure is non-trivial in the sense that it is not reducible to a multisemigroup structure. The above formulae determine the multiplicity functions µ s,w and µ t,w . As any element in D n is a product of s and t, all remaining multiplicity functions can be determined inductively. However, we do not know of any explicit formulae. For n = 3, the answer is given in (2). More information on the D n case can be found in [El].
Let C be a small category equivalent to A-mod. Following [GM1], we define the 2-category G n as a subcategory in Cat (not full) such that: • C A has one object i, which we identify with C; • 1-morphisms in C A are functors isomorphic to direct sums of functors of tensoring with subbimodules of A A A ; • 2-morphisms in C A are natural transformations of functors.
The main result of [GM1] asserts that the multisemigroup S(G n ) (with added zero) is isomorphic to the Catalan monoid C n+1 of all order-preserving and orderdecreasing transformation of a finite chain with n + 1 elements. In particular, the multisemigroup S(G n ) is a semigroup.
Moreover, in [GM1] it is also shown that the composition of indecomposable 1morphism in G n is indecomposable (or zero). This means that, in this case, the multi-multisemigroup structure on S(G n ) coincides with the multisemigroup structure.
A similar phenomenon was observed in some other cases in [Zh1,Zh2].
Proof. The axiom (4) in the new situation (for λ) follows from the axiom (4) in the old situation (for κ) by applying the homomorphism Φ λ,κ of complete semirings to both sides.
A special case of this construction was mentioned in Subsection 4.1, in that case κ = ω and λ = 1. A natural question is whether this construction is "surjective" in the sense that any multisemigroups with multiplicities bounded by λ can be obtained in this way from a multisemigroups with multiplicities bounded by κ. If λ = 1, the answer is yes due to the following construction: Let κ be a nonzero cardinal numbers. Then there is a canonical homomorphism Ψ κ : Card 1 ∼ = B → Card κ which sends 0 to 0 and also sends 1 to κ. Given a multisemigroup (S, * ), we thus may define µ s,t (r) := 0, r ∈ s * t; κ, r ∈ s * t. In other words, we define µ as the composition of * followed by Ψ κ . Similarly to the proof of Proposition 11 we thus get that (S, µ) is a multisemigroups with multiplicities bounded by κ. As the homomorphism Φ 1,κ • Ψ κ is the identity on B, we obtain (S, * ) = (S, Φ 1,κ • µ).
s6.2 6.2. Finitary cardinal reduction. To avoid degenerate examples above, it is natural to rephrase the question as follows: Given a multisemigroup (S, * ), whether there is a finitary multi-multisemigroup (S, µ) such that (S, * ) = (S, Φ 1,ω • µ). The following example shows that this is, in general, not the case. Proof. First we show that the multiplication table (10) really defines a multisemigroup. We need to check the associativity axiom x * (y * z) = (x * y) * z. If x = y = z = a, then both sides are equal to a. If x = b or y = b or z = b, then both sides are equal to {a, b}.
Now assume that ({a, b}, µ) is a finitary multisemigroup with multiplicities. Then µ a,b (a) = 0 because a ∈ a * b, moreover, we have µ a,b (b) = 0 as b ∈ a * b.
We want to compute µ aab (a) in two different ways, namely, using the decompositions (aa)b and a(ab). In the first case, we get µ aab (a) = µ a,a (a)µ a,b (a). In the second case, we obtain µ aab (a) = µ a,a (a)µ a,b (a) + µ a,b (b)µ a,b (a). As both µ a,b (a) = 0 and µ a,b (b) = 0, we get a contradiction. The claim follows.
s6.3 6.3. Deformation of multisemigroups. Let (S, * ) be a finite multisemigroup. A finitary multi-multisemigroup (S, µ) such that (S, * ) = (S, Φ 1,ω • µ) is called a deformation of (S, * ). As we saw above, not every finite multisemigroup admits a deformation. It would be interesting to find some sufficient and necessary conditions for a multisemigroup to admit a non-trivial deformation. Ideally, it would be really interesting to find some way to describe all possible deformations of a given multisemigroup. The following is a corollary from the result in the previous subsection.

cor32
Corollary 13. Let (S, * ) be a multisemigroup containing two different elements a and b such that a * a = {a} and {a, b} ⊂ a * b or {a, b} ⊂ b * a. Then (S, * ) does not admit any deformation.
Proof. In the case {a, b} ⊂ a * b, the claim follows from the arguments in the proof of Proposition 12. In case {a, b} ⊂ b * a the proof is similar.
Another obvious observation is the following.
Proof. By construction, (S(C ), µ) is a deformation of (S(C ), * ). s6.4 6.4. Connection to twisted semigroup algebras. In case a finite multisemigroup (S, * ) is a semigroup, deformations of (S, * ) can be understood as integral twisted semigroup algebras in the sense of [GX]. Indeed, according to the above definition, a deformation of a finite semigroup (S, * ) is given by a map which satisfies axiom (4) (the associativity axiom). This is a special case of the definition of twisted semigroup algebras, see [GX,Section 3] or [Wi,Equation (1)]. Typical examples of semigroups which admit non-trivial twisted semigroup algebras (and hence also non-trivial deformations) are various diagram algebras, see [MaM, Wi] for details.

7.
Multi-multisemigroups and modules over complete semirings s7 s7.1 7.1. Modules over semirings. Let R be a unital semiring. A (left) R-module is a commutative monoid (M, +, 0) together with the map · : R × M → M , written (r, m) → r · m, such that • (rs) · m = r · (s · m), for all r, s ∈ R and m ∈ M ; • (r + s) · m = r · m + s · m, for all r, s ∈ R and m ∈ M ; • r · (m + n) = r · m + r · n, for all r ∈ R and m, n ∈ M ; • 0 · m = 0, for all m ∈ M ; • 1 · m = m, for all m ∈ M . We refer, for example, to [JM] for more details. For instance, the multiplication on R defines on R the structure of a left R-module R R, called the regular module.
s7.2 7.2. Modules over complete semirings. Let R be a complete unital semiring. A (left) complete R-module is an R-module (M, +, 0, ·) such that • M is complete; • r · i∈I m i = i∈I r · m i , for all r ∈ R and m i ∈ M ; • i∈I r i · m = i∈I r i · m, for all r i ∈ R and m ∈ M .
For example, the regular R-module is complete. Another important for us example of a complete R-module is the following. Example 15. Let R be a complete unital semiring and X a non-empty set. Then the set R X of all functions from X to R is a complete abelian group with respect to component-wise addition, moreover, it has the natural structure of a complete Rmodule given by component-wise multiplication with elements in R. This module has, as an incomplete submodule, the set of all functions in R X with finitely many non-zero values. 7.3. Algebras over complete semirings. For a complete unital semiring R and a non-empty set X, consider the complete R-module R X as in Example 15 above. An algebra structure on R X is a map • : R X × R X → R X such that, for all f i , f, g, h ∈ R X , we have i∈I For example, if X = {a}, then R X = R and the multiplication · on R defines on R the structure of a complete R-algebra.
s7.4 7.4. Connection to multi-multisemigroups. If R is a semiring and X a set, then, for x ∈ X, we denote by χ x : X → R the indicator function of x defined as follows: Our main result in the section is the following: thm31 Theorem 16.
thm31.1 (i) Let κ be a cardinal and (S, µ) be a multisemigroup with multiplicities bounded by κ. Then Card S κ has a unique structure of a complete Card κ -algebra such that χ s • χ t = µ s,t , for all s, t ∈ S. thm31.2 (ii) Conversely, any complete Card S κ -algebra (Card S κ , •) defines a unique structure of a multisemigroup with multiplicities bounded by κ on S via µ s,t := χ s • χ t , for s, t ∈ S.
Proof. To prove claim (i), we first note that uniqueness would follow directly from existence as any element in Card S κ can be written as a sum, over S, of indicator functions. To prove existence, we note that each function can be uniquely written as a sum, over S, of indicator functions. Therefore, there is a unique way to extend χ s • χ t := µ s,t , for s, t ∈ S, to a map • : Card S κ × Card S κ → Card S κ which satisfies (11) and (12). Using (11) and (12), it is enough to check the axiom (13) for the indicator functions, where it is equivalent to the axiom (4), by definition. This proves claim (i).
To prove claim (ii), we only need to check the axiom (4). This axiom follows from the axiom (13) applied to the indicator functions. This completes the proof.
Theorem 16 suggests that one could define multisemigroups with multiplicities from any complete semiring, not necessarily Card κ .