Ideals of Largest Weight in Constructions Based on Directed Graphs

We introduce a new construction based on directed graphs. It provides a common generalization of the incidence rings andMunn semirings. Our main theorem describes all ideals of the largest possible weight in this construction. Several previous results can be obtained as corollaries to our new main theorem. Introduction Many interesting results have been obtained recently in the literature devoted to properties of graphs (cf. [12, 13, 14, 15, 16, 21, 28, 33]) as well as algebraic constructions (cf. [22, 25, 27, 29, 31]). Incidence algebras of directed graphs have been studied for a long time (cf. [24, 31]). On the other hand, Munn semirings have also been considered by several authors too (cf. [9]). They were defined by analogy with the Munn algebras, which are very well known (cf. [30]). The present paper introduces Munn incidence semirings as a common generalization of the incidence algebras and Munn semirings. A complete definition of this new construction is given in the section “Main Results” below. The main theorem of this paper is Theorem 1 in the same section. It describes all ideals of the largest possible weight in Munn incidence semirings. This result belongs to the large area studying the weights of ideals in various constructions. For instance, the ideals of the largest possible weight have been described for certain classes of the polynomial quotient rings in [26], structural matrix semirings in [17], Munn semirings in [9], and incidence semirings in [8, 10]. The investigation of ideals with largest weight has been motivated by applications of ideals in data mining (cf. [8, 11, 25]), health informatics (cf. [4, 6, 7, 19, 20, 32]) and information security (cf. [1, 2, 5]). It is important to describe the weights of ideals in the new and more general construction of the Munn incidence semirings for the following reasons. First, the ideals in a more general construction may lead to a discovery of systems with better properties crucial for cryptographic applications. Second, it is nice to unify already known facts in such a way that several previous results can be derived as corollaries to a new and more general theorem. In particular, proofs of the main results of the previous papers [8], and [10] can now be derived from the main theorem of the present article.

The present paper introduces Munn incidence semirings as a common generalization of the incidence algebras and Munn semirings. A complete definition of this new construction is given in the section "Main Results" below. The main theorem of this paper is Theorem 1 in the same section. It describes all ideals of the largest possible weight in Munn incidence semirings. This result belongs to the large area studying the weights of ideals in various constructions. For instance, the ideals of the largest possible weight have been described for certain classes of the polynomial quotient rings in [26], structural matrix semirings in [17], Munn semirings in [9], and incidence semirings in [8,10]. The investigation of ideals with largest weight has been motivated by applications of ideals in data mining (cf. [8,11,25]), health informatics (cf. [4,6,7,19,20,32]) and information security (cf. [1,2,5]).
It is important to describe the weights of ideals in the new and more general construction of the Munn incidence semirings for the following reasons. First, the ideals in a more general construction may lead to a discovery of systems with better properties crucial for cryptographic applications. Second, it is nice to unify already known facts in such a way that several previous results can be derived as corollaries to a new and more general theorem. In particular, proofs of the main results of the previous papers [8], and [10] can now be derived from the main theorem of the present article.

Preliminaries
We use standard terminology and refer the readers to [4,13,18,22,23,31] for more detailed explanations of notions used in the present article. Throughout the word 'digraph' means a directed graph without multiple parallel edges but possibly with loops, and D = (V, E) is a digraph with the set V of vertices and the set E of edges. The set of all positive integers is denoted by N.
Here we use the standard definition of a semiring without assuming that every semiring contains an identity element, see [9] and [17].
For any subset S of a semiring Q, the ideal generated by S in Q is denoted by id(S) and is defined as the set of all sums of these elements and their multiples, i.e., It is easily seen that the set id(S) is a subsemiring of Q closed for the multiplication by the elements of Q, i.e., id(S)Q + Q id(S) ⊆ id(S).
Recall that a semiring Q is said to be idempotent if x + x = x for all x ∈ Q. A semiring Q is said to be zero-divisor-free, if xy = 0 implies x = 0 or y = 0, for any x, y ∈ Q.
Let Q be a semiring, I and Λ finite nonempty sets, and let P be a Λ × I-matrix with entries in Q. A Munn semiring over Q with sandwich-matrix P is the set M (Q; I, Λ; P ), consisting of all I × Λ matrices over Q, equipped with the usual addition and multiplication · defined by A · B = AP B, for A, B ∈ M (Q; I, Λ; P ).
The concept of an incidence semiring was defined in [8] as an exact analogue of an incidence algebra. Let Q be a semiring. The incidence algebra of D = (V, E) over Q is denoted by I D (Q) and is defined as the set consisting of zero 0 and all finite sums endowed with the usual addition and with multiplication defined by the distributive law and the rule for all (u 1 , v 1 ), (u 2 , v 2 ) ∈ E, see [22,31].

Main Results
Let D = (V, E) be a digraph with the (possibly infinite) set V of vertices and set E of edges, and let I and Λ be nonempty sets such that V = I ∪ Λ. Let Q be a semiring, and let P = [p λi ] be a (Λ × I)-matrix with entries p λi ∈ Q, for all λ ∈ Λ, i ∈ I. For i ∈ I, λ ∈ Λ, denote by e iλ the standard elementary I × Λ matrix with 1 in the intersection of i-th row and λ-th column and zeros in all other entries. Denote by M D (Q; I, Λ; P ) the set of all I × Λ matrices x of the form where (i j , λ j ) ∈ E ∩ (I × Λ) and 0 ̸ = x i j λ j ∈ Q, for j = 1, . . . , n. Endow this set with the usual componentwise addition of matrices and with a multiplication · defined by the distributive laws and the rule for any (i 1 , λ 1 ), (i 2 , λ 2 ) ∈ E ∩ (I × Λ) and any x i 1 λ 1 , x i 2 λ 2 ∈ Q. If the multiplication defined by (4) is associative, then we say that M D (Q; I, Λ; P ) is a Munn incidence semiring of the digraph D. Lemmas 2 and 3 in Section show that Munn incidence semirings of digraphs are a common generalisation of the standard Munn semirings and incidence algebras, including the incidence algebras of infinite digraphs. Let M D (Q; I, Λ; P ) be a Munn incidence semiring of the digraph D = (V, E). For each element x in M D (Q; I, Λ; P ), the weight wt(x) of x is equal to the number of nonzero coefficients x i j λ j in the finite sum (3) for x (cf. [23, §2.13]). If S is a subset of M D (Q; I, Λ; P ), then the weight of S is denoted by wt(S) and is defined as the minimum weight of a nonzero element in S.
For any subset S of F = M D (Q; I, Λ; P ) and any i ∈ I, λ ∈ Λ, A ⊆ E, X ⊆ I, Y ⊆ Λ, we use the following notation: We introduce the following sets of edges For any vertex v ∈ V , define the following sets of vertices: Further, we assume that D = (V, G) is a finite digraph. Then the cardinality N Z = |E R ∩ E L | is an integer. Besides, denote by N L the largest positive integer such that the set K L (N L ) is not empty, and put N L = 0 if such integers do not exist. Likewise, denote by N R the largest positive integer such that the set K F (N R ) is not empty, and put N R = 0 if such integers do not exist.
Let us define the following subsets of the Munn incidence semiring M D (Q; I, Λ; P ): Our main theorem describes ideals that have the largest possible weight among the weights of all ideals in Munn incidence semirings. x ∈ T ;

Technical Lemmas and Proofs
Let us begin with a few technical properties of the Munn incidence semirings. The following two lemmas show that Munn incidence semirings of digraphs are a common generalization of the incidence algebras and Munn semirings.
Proof follows immediately from the definitions of the Munn incidence semiring and incidence algebra. Proof follows immediately from the definitions of the Munn incidence semiring and classical Munn semiring.
We say that the digraph D is P -balanced if, for all i 1 , λ 1 and p λ 1 i 2 , p λ 2 i 3 ̸ = 0, the following equivalence holds: Proof. Suppose that the digraph D is P -balanced. Then we are going to verify that the associative law x(yz) = (xy)z is satisfied, for all x, y, z ∈ M D (Q; I, Λ; P ). The distributive law and equality (3) imply that it is enough to consider the case where x = q x e ixλx , y = q y e iyλy , z = q z e izλz , for some q x , q y , q z ∈ Q and (i x , λ x ), (i y , λ y ), (i z , λ z ) ∈ E ∩ (I × Λ).
If (i x , λ y ) / ∈ E, then (4) tells us that xy = 0, and so (xy)z = 0. Since D is P -balanced, we get (i y , λ z ) / ∈ E. Therefore yz = 0 and x(yz) = 0, and so the associative law holds. It remains to consider the case where (i x , λ y ) ∈ E. Then (i y , λ z ) ∈ E, because D is P -balanced. Therefore we get (xy)z = (q x p λxiy q y )e ixλy · z = (q x p λxiy q y p λyiz q z )e ixλz = x · (q y p λyiz q z )e iyλz = x(yz). Thus, the associative law holds true, as required.
Conversely, let us assume that M D (Q; I, Λ; P ) is a semiring. Then we have to verify that the equivalence (14) holds. By way of contradiction, suppose that there exist i x , λ x , i y , λ y , i z , λ z such that (i x , λ x ), (i y , λ y ), (i z , λ z ), (i x , λ z ) ∈ E ∩ (I × Λ) and p λxiy , p λyiz ̸ = 0.
The following lemma is easy and well known.
For any semiring Q, the left annihilator of Q is the set Ann L (Q) = {x ∈ Q | xQ = 0}, and the right annihilator of Q is the set Ann R (Q) = {x ∈ Q | Qx = 0}. It follows immediately that Ann L (Q) and Ann R (Q) are ideals of the semiring Q.

Lemma 6. Let Q be a zero-divisor-free idempotent semiring with identity element, and let F be a Munn incidence semiring M D (Q; I, Λ; P ) of the digraph D = (V, E). Then Ann R (F ) = F E L and
Proof. Put T R = Ann R (F ). First, let us prove the inclusion T R ⊇ F E L . Suppose to the contrary that there exists a nonzero element x in F E L such that x / ∈ T R . Then we have F x ̸ = 0, and so there exists qe iλ ∈ F such that qe iλ x ̸ = 0. The definition of F E L and equality (3) show that x = ∑ n j=1 q j e i j λ j , for some n > 0, 0 ̸ = q j ∈ Q, q j ∈ Q, (i j , λ j ) ∈ E L . It follows that qe iλ q j e i j λ j ̸ = 0, for some j. Hence (4) yields that p λi j ̸ = 0 and (i, λ j ) ∈ E. Therefore (5) implies that (i j , λ j ) / ∈ E L . This contradicts the choice of x in F E L and shows that x belongs to T R . Thus, Second, let us prove the reversed inclusion. Choose any element x in T R . Equality (3) implies that Then the definition of F E L shows that there exists j such that (i j , λ j ) belongs to E \ E L . It follows from (5) that there exists (w, µ) in E such that (w, λ j ) ∈ E and p µi j ̸ = 0. Therefore (4) yields that e wµ (q j e i j λ j ) = (p µi j q j )e w,λ j ̸ = 0 in F , where p µi j q j ̸ = 0 because Q is zero-divisor-free. However, e wµ q j e i j λ j is a summand of e wµ x with coefficient q j . Since Q is an idempotent semiring, Lemma 5 shows that this summand does not cancel with other summands of e wµ x. Therefore e wµ x ̸ = 0. This contradicts the choice of x in T R , and shows that x belongs to F E L .
Thus, T R = F E L , as required. The proof of the second equality Ann L (F ) = F E R is dual and we omit it. To prove that wt( id(x)) = wt(x), choose a nonzero element y in id(x). The inequality wt( id(x)) ≤ wt(x) is obvious, and so it remains to verify that wt(y) ≥ wt(x).

Lemma 7. Let Q be a zero-divisor-free idempotent semiring with identity element, and let F be a Munn incidence semiring
It follows from (1) that y can be written down in the form y = ∑ k j=1 ℓ j xr j , for some ℓ j , r j ∈ F ∪ {1}. We may assume that only nonzero summands ℓ j xr j have been included in representation for y given above. The definition of K L (N L ) tells us that (i, λ) ∈ E L , for all i ∈ Y . Lemma 6 shows that x belongs to Ann R (F ). This forces all products ℓ j x to be equal to zero whenever ℓ j is in F . It follows that ℓ 1 = · · · = ℓ k = 1.
In view of (3) each element r j ∈ F , that occurs in the expression for y given above, can be represented in the form r j = ∑ b a=1 h j (a)e i j (a)λ j (a) , for some b ∈ N, h j (a) ∈ Q, i j (a) ∈ I, λ j (a) ∈ Λ. Substituting these representations in the sum for y given above and applying the distributive law, to simplify notation we may assume that from the very beginning each element r j ̸ = 1 itself has the form Given that x ∈ A L and (Y, λ) ∈ K L (N L ), the last condition in the definition of K L (N L ) tells us that the intersection Out(i) ∩ Out P (λ) is equal to one and the same set for all i ∈ Y , i.e., Out(i 1 ) ∩ Out P (λ) = Out(i 2 ) ∩ Out P (λ), for all i 1 , i 2 ∈ Y . Since xr j ̸ = 0, it follows from (4) that p λi j ̸ = 0 and that there exists i 1 in Y such that λ j belongs to Out(i 1 ) ∩ Out P (λ). Consequently, λ j is in Out(i) ∩ Out P (λ), for all i ∈ Y .
Fix any i ∈ Y and consider the summand q i e iλ of x. Since Q is zero-divisor-free and p λi j ̸ = 0, we get q i p λi j h j ̸ = 0. Hence (4), (9) and (10) imply that q i e iλ · r j = q i e iλ · h j e i j λ j = (q i p λi j h j )e iλ j ̸ = 0.
It follows that wt(xr j ) = |Y | = wt(x), for each r j ̸ = 1. The equality wt(xr j ) = wt(x) also holds trivially for each r j = 1 in the sum for y. Since Q is an idempotent semiring, it follows from Lemma 5 that wt(y) ≥ wt(x). By the choice of x, this means that wt( id(x)) = wt(x). This completes the proof.

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BMSA Volume 15 Proof. Take any element x in A Z . By (13), it can be recorded in the form Choose a nonzero element y in id(x). It follows from (1) that y can be written down as y = ∑ k j=1 ℓ j xr j , for some ℓ j , r j ∈ F ∪ {1}. Since Q is an idempotent semiring with identity element, Lemma 6 tells us that Ann It follows that if ℓ j ∈ F or r j ∈ F , then ℓ j xr j = 0 in the representation for y. Therefore we may assume that ℓ j = r j = 1, for all summands of the representation of y given above. This means that id(x) = Nx. Now, take any element kx ̸ = 0 in Nx, where k ∈ N. Suppose that kq iλ = 0 for some summand q iλ e iλ of x. Since Q has an identity element 1 Q , we infer that k1 Q = kx ̸ = 0 in Q. The equalities (k1 Q )q iλ = kq iλ = 0 mean that k1 Q is a zero divisor in Q. This contradicts the hypothesis that Q is zero-divisor-free. It follows that, for every k ∈ N such that kx ̸ = 0, the weight of kx is equal to wt(x). We conclude that wt( id(x)) = wt(x), which completes the proof.
Proof of Theorem 1. To prove condition (a) we suppose that wt(T ) > 1. Choose a nonzero element x of minimal weight in T . Then we have wt(x) = wt(T ) > 1. Lemma 6 says that Ann L (F ) = F E R and Ann R (F ) = F E L .
If x / ∈ Ann R (F ) ∪ Ann L (F ), then there exist q 1 e i 1 λ 1 and q 2 e i 2 λ 2 in F such that the products q 1 e i 1 λ 1 x and xq 2 e i 2 λ 2 are nonzero. Equality (4) yields us that the product y = q 1 e i 1 λ 1 xq 2 e i 2 λ 2 is nonzero too. Since Q is zero-divisor-free, it also follows from (4) that y ∈ F i 1 λ 2 ; whence wt(y) = 1. However, y belongs to T . This contradicts the hypothesis that wt(T ) = 1, and shows that x always belongs to the union Ann R (F ) ∪ Ann L (F ). Therefore the following three cases may occur.
and so wt(T ) = wt(x) ≤ N Z . On the other hand, it follows from the maximality of wt(T ) and Lemma 9 that wt(T ) ≥ N Z . Therefore wt(x) = N Z . Hence (13) shows that x ∈ A Z , which means that condition (a) holds. Case 2.
x ∈ Ann R (F ) \ Ann L (F ). Then x ∈ F E L . In view of (3), we infer where n ≥ 0, 0 ̸ = q j ∈ Q, and (i j , λ j ) ∈ E L . Since x does not belong to Ann L (F ), it follows that there exists qe iλ ∈ F such that xqe iλ ̸ = 0. Consider the set Y = {i 1 , . . . , i n }. We are going to verify that the pair (Y, λ) satisfies all conditions in the definition of K L (n).
Third, given that x belongs to the ideal Ann R (F ) of F , it is clear that xqe iλ is in Ann R (F ), too. Hence (i j , λ) ∈ E L , for all i j ∈ Y , by Lemma 6.
Fourth, it is clear that |Y | = n, as required in the definition of K L (n), too.
Fifth, to prove the last property required in the definition of K L (n), suppose to the contrary that the intersections Out(i 1 ) ∩ Out P (λ) and Out(i 2 ) ∩ Out P (λ) are different for some i 1 , i 2 in Y . Without loss of generality we may assume that there exists µ ∈ Λ that belongs to Out(i 1 ) ∩ Out P (λ) but does not belong to Out(i 2 ). Then (i 1 , µ) ∈ E and (i 2 , µ) / ∈ E. Since µ ∈ Out P (λ), by (10) there exists j ∈ I such that p λj ̸ = 0 and (j, µ) ∈ E. Since Q is a semiring with identity element, we have e jµ ∈ F . Therefore (4) implies that q 1 e i 1 λ e jµ ̸ = 0, but q 1 e i 1 λ e jµ = 0. It follows that wt(xe jµ ) < wt(x). This contradicts the minimality of wt(x) in T , because xe jµ ∈ T . The contradiction shows that all intersections Out(i j ) ∩ Out P (λ) are equal to one and the same set, for all i j ∈ Y . Thus, we have proved that (Y, λ) belongs to K L (n).
Further, the maximality of N L implies that |Y | ≤ N L . It follows from (15) that xqe iλ ∈ F Y λ . Hence we get wt(x) = wt(xqe iλ ) ≤ |Y | ≤ N L . On the other hand, the maximality of wt(T ) and Lemma 7 show that wt(T ) ≥ N L . Hence wt(x) ≥ N L . Therefore wt(xqe iλ ) = wt(x) = N L , and so |Y | = N L . Thus, we conclude that n = wt(x) = N L and (Y, v) ∈ K L (N L ). By (11), we get x(g; i, λ) ∈ A L . Therefore condition (a) holds in this case too.
x ∈ Ann L (F ) \ Ann R (F ). Then the proof is dual to the proof of Case 2, and so condition (a) holds again. This concludes the proof of condition (a).
To verify condition (b), first note that the inequality wt(T ) ≥ max{N L , N R , N Z } follows immediately from Lemmas 7, 8 and 9 and the maximality of wt(T ). If wt(T ) = 1, then we get N L , N R , N Z ≤ 1; whence condition (b) is met. On the other hand, if wt(T ) > 1, then condition (a) implies that wt(T ) ≤ max{N L , N R , N Z }, and so (b) holds again. This completes the proof.