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We introduce a generalization of the universal valuation semiring defined by Jeffrey and Noah Giansiracusa. We then explicitly characterize the additive structure of this semiring and show that, when applied to Q , this characterization gives the non-Archimedean case of Ostrowski's theorem. We conclude with examples of non-commutative valuations and their applications, such as the detection of the existence of representations of rings in ultrametric vector spaces. [ABSTRACT FROM AUTHOR]

This paper introduced a novel semiring structure involving nonnegative integers, where operations depended on the comparison of the magnitudes of decimal digit sums. Consequently, a corresponding matrix semiring can be established on this commutative semiring. We showed that the 3-satisfiability problem can be polynomial-time reduced to solving systems of quadratic polynomial equations over this semiring. We proposed a key exchange protocol based on this matrix semiring, with its security relying on the two-sided digital circulant matrix action problem over this semiring. This scheme provides a novel cryptographic primitive for post-quantum cryptography.

Based on the minimal and simple representations of semirings, we introduce two Jacobson-type Hoehnke radicals, namely, m-radical and s-radical, of a semiring S. Every minimal (simple) S-semimodule is a quotient of S by a regular right congruence (maximal) μ on S such that [0]μ is a maximal μ-saturated right ideal in S. Thus the m(s)-radical becomes an intersection of a class of regular congruences. The m(s)-radical of the direct product of two semirings is the product of their individual m(s)-radicals. [ABSTRACT FROM AUTHOR]

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c ∈ (0 , 1 ] such that I = (− x , y) ∈ R 2 | c x ≤ y ≤ x c and x , y ≥ 0 . This shows that its shape has to be symmetric with the graph y = − x . [ABSTRACT FROM AUTHOR]

In the present paper, we study the connection between the commutativity of a WLC semiring and derivations. Moreover, we establish a complete description and classification for some of these derivations. Finally, we give an example to prove that the imposed hypotheses are necessary. [ABSTRACT FROM AUTHOR]

Let R be a commutative semiring and I be a k-ideal of R. In this paper, we introduce the k-ideal-based graph of R, denoted by ΓI ∗ (R). The basic properties and possible structures of the graph are studied. [ABSTRACT FROM AUTHOR]

We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is commutative, if every idempotent in the semiring has an orthogonal complement and all the nilpotents in the semirings are central. We also provide examples that the assumptions on the existence of orthogonal complements of idempotents and the centrality of nilpotents cannot be omitted. [ABSTRACT FROM AUTHOR]

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring M n (S) , of all n × n matrices over a semiring S, is congruence-simple, provided that either S has a multiplicatively absorbing element or S is commutative and additively cancellative. [ABSTRACT FROM AUTHOR]

We show that the additively idempotent semiring S 7 0 has no finite basis for its equational theory. This answers an open problem posed by Jackson et al. (J Algebra 611:211–245, 2022). [ABSTRACT FROM AUTHOR]

In this paper, Rad-⨁-supplemented semimodules are defined as generalization of⨁-supplemented semimodules. Let R be a semiring. An R-semimodule A is called a Rad-⨁-supplemented semimodule, if each subsemimodule of A has a Rad-supplement which is a direct summand of A. Here, we investigate some properties of these semimodules and generalize some results on Rad-⨁-supplemented modules to semimodules. We prove that any finite direct sum of Rad-⨁-supplemented semimodules is Rad-⨁-supplemented. Also, we prove that if A is a subtractive semimodule with (D3) then A is Rad-⨁-supplemented if and only if every direct summand to A is Rad-⨁-supplemented. [ABSTRACT FROM AUTHOR]

We first obtain several subdirect product decompositions of additively regular semirings by constructing several pairs of generalized factor congruences. Next, we define and study several additively regular semiring varieties by congruence openings and closure congruences. Moreover, as an application, we obtain several pullback product decompositions of additively regular band semirings. Finally, we establish order embedding theorems on the poset of subvarieties of the additively regular semiring e-variety A R . [ABSTRACT FROM AUTHOR]

We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory, (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case. [ABSTRACT FROM AUTHOR]

Green's ∗-relations on semirings satisfying additional identities x n ≈ x and 2 (x + y) ≈ 2 x + 2 y are studied. Equivalent characterizations of ℋ ∗ + ∧ ℒ ∗ • , ℋ ∗ + ∧ ℛ ∗ • , ℋ ∗ + ∧ ∗ • , ℋ ∗ + ∧ ℋ ∗ • , ℋ ∗ + ∨ ℒ ∗ • , ℋ ∗ + ∨ ℛ ∗ • , ℋ ∗ + ∨ ∗ • and ℋ ∗ + ∨ ℋ ∗ • are obtained and equivalent conditions are given for the above relations to be congruences. It is proved that semiring classes determined by these congruences are varieties. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of liner fuzzy real numbers, and show that the set of all positive (or negative) symmetric linear fuzzy real numbers forms a semiring. Moreover, we discuss a complex transform of linear fuzzy real numbers. [ABSTRACT FROM AUTHOR]

Let M be a semimodule over a commutative semiring R and K be a subtractive subsemimodule of M with K* = K \ {0}. The subtractive subsemimodule-based graph of M is defined as the simple undirected graph Ω = ΓK* (M) with vertex set V (Ω) = {v ∈ M \ K : v + v′ ∈ K* for some v ̸= v′ ∈ M \ K}, and two distinct vertices m and n are adjacent if and only if m + n ∈ K*. In this paper, we study the interplay between semimodule properties and the properties of the graph. Among other results, we compute the diameter and the girth of ΓK* (M). [ABSTRACT FROM AUTHOR]

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c∈(0,1] such that I=(−x,y)∈R2|cx≤y≤xcandx,y≥0. This shows that its shape has to be symmetric with the graph y=−x.

Assumed that (S,+,.) is a semiring. Semiring is a algebra structure as a generalization of a ring. A set I⊆S is called an ideal over semiring S if for any α,β∈I, we have α-β∈I and sα=αs∈I for every s in semiring S. Based on this definition, there is a special condition namely prime ideal P, when for any αβ∈P, then we could prove that α or β are elements of ideal P. Furthermore, an ideal I of S is irreducible if Ia is an intersection ideal from any ideal A and B on S, then I=A or I=B. We also know the strongly notion of the irreducible concept. The ideal I of S is a strongly irreducible ideal when I is a subset of the intersection of A and B (ideal of S), then I is a subset of A, or I is a subset of B. In this paper, we discussed the characteristics of the semiring of the non-negative integer set. We showed that pZ^+ is an ideal of semiring of the non-negative integer Z^+ over addition and multiplication. We find a characteristic that 〖pZ〗^+ is a prime ideal and also a strongly irreducible ideal of the semiring Z^+ with p is a prime number.

Green’s ∗-relations on semirings satisfying additional identities [Formula: see text] and [Formula: see text] are studied. Equivalent characterizations of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are obtained and equivalent conditions are given for the above relations to be congruences. It is proved that semiring classes determined by these congruences are varieties.

We find the metric dimension of the zero-divisor graph of the matrix semiring over a commutative entire antinegative semiring. [ABSTRACT FROM AUTHOR]

Summand absorbing submodules are common in modules over (additively) idempotent semirings, for example, in tropical algebra. A submodule W of V is summand absorbing, if x + y ∈ W implies x ∈ W,y ∈ W for any x,y ∈ V. This paper proceeds the study of these submodules, and more generally of additive monoids, with emphasis on their archimedean classes and quotient structures. [ABSTRACT FROM AUTHOR]

Proto-exact categories , introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B -modules as proto-exact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfield K , a well-known relation between finite K -modules and finite incidence geometries yields a combinatorial interpretation of exact sequences. [ABSTRACT FROM AUTHOR]

Let S be an additively idempotent semiring and Mn(S) be the semiring of all n×n matrices over S. We characterize the conditions of when the semiring Mn(S) is congruence-simple provided that the semiring S is either commutative or finite. We also give a characterization of when the semiring Mn(S) is subdirectly irreducible for S being almost integral (i.e. xy + yx + x = x for all x,y ∈ S). In particular, we provide this characterization for the semirings S derived from the pseudo MV-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we provide a classification of congruence-simple semirings with a multiplicatively absorbing element and without nontrivial nilpotent elements. [ABSTRACT FROM AUTHOR]

A weighted finite automaton has the finite-image property if the image of the weighted language associated with it is finite. We show two undecidability results concerning the finite-image property of weighted finite automata over semirings, respectively strong bimonoids. Firstly, we give a computable idempotent commutative past-finite ordered semiring such that it is undecidable, for an arbitrary deterministic weighted finite automaton over that semiring, whether has the finite-image property. Secondly, we give a computable commutative past-finite monotonic ordered strong bimonoid such that it is undecidable, for an arbitrary weighted finite automaton over that strong bimonoid, whether has the finite-image property. This shows that recent decidability results for suitable weighted finite automata over past-finite monotonic strong bimonoids cannot be extended to natural classes of ordered semirings and ordered strong bimonoids without further assumptions. [ABSTRACT FROM AUTHOR]

The term semidomain refers to a subset S of an integral domain R, in which the pairs (S , +) and (S , ·) are semigroups with identities. If S contains no additive inverses except 0, we say that S is additively reduced. By taking polynomial expressions with coefficients in S and exponents in a torsion-free monoid M, we obtain the additively reduced monoid semidomain S[M]. In this paper, we investigate the factorization properties of such semidomains, providing necessary and sufficient conditions for them to be bounded factorization semidomains, finite factorization semidomains, and unique factorization semidomains. We also identify large classes of semidomains with full and infinite elasticity. Throughout the paper, we present examples to help elucidate the arithmetic of additively reduced monoid semidomains. [ABSTRACT FROM AUTHOR]

Fuzzy sets, soft sets, and their generalisations have always been important tools for mathematicians and researchers working with uncertainty. Jun proposed a hybrid structure that combined the concepts of a fuzzy set and a soft set. It should be noted that hybrid structures are a combination of soft set and fuzzy set speculation. Our aim is to explore the concept of hybrid ordered ideals and hybrid interior ideals in ordered semirings, as well as look at some of their related properties, which is one of the important aspects of this paper. In order to investigate the structure theory of hybrid ideals in ordered semirings, we define hybrid composition and hybrid addition. We also establish and characterise the regularity of ordered semirings in terms of hybrid structures. [ABSTRACT FROM AUTHOR]

In the present paper, we study some of the topological properties of the space of prime congruences on a semiring endowed with the hull kernel topology. [ABSTRACT FROM AUTHOR]

We prove that each element of a complete atomic -semiring has a canonical decomposition. We also find some sufficient conditions for the decomposition to be unique that are expressed by first-order sentences. As a corollary, we obtain a theorem of Avgustinovich–Frid which claims that each factorial language has the unique canonical decomposition. [ABSTRACT FROM AUTHOR]

We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities. [ABSTRACT FROM AUTHOR]

We study left pure ideals, left pure k-ideals, right pure ideals and right pure k-ideals of semirings, introduce the notions of quasi-pure ideals, quasi-pure k-ideals, bi-pure ideals and bi-pure k-ideals of semirings and characterize regular and k-regular semirings using their pure ideals and their pure k-ideals, respectively. [ABSTRACT FROM AUTHOR]

In this paper‎, ‎as a further generalization of fuzzy ideals‎, ‎we introduce the notion of a fuzzy (soft) quasi-interior ideals of semirings and characterize regular semiring in terms of fuzzy (soft) quasi-interior ideals of semirings‎. ‎We prove that $(\mu‎, ‎A)$ is a fuzzy soft left quasi-interior ideal‎ ‎over a regular semiring $M,$ if and only if $(\mu‎, ‎A)$ is a fuzzy soft quasi-ideal over a semiring $M,$‎ ‎and study some of the properties‎.

In this paper, we consider Leavitt path algebras having coefficients in a k-semifield. Concentrating on the aspect of k-simplicity, we find a set of necessary and sufficient conditions for the k-simplicity of the Leavitt path algebra LS(Γ) of a directed graph Γ over a non-zeroid k-semifield S.

Let S be a multiplicatively idempotent congruence-simple semiring. We show that | S | = 2 if S has a multiplicatively absorbing element. We also prove that if S is finite then either | S | = 2 or S ≅ End (L) or S op ≅ End (L) where L is the 2-element semilattice. It seems to be an open question, whether S can be infinite at all. [ABSTRACT FROM AUTHOR]

Semiring is an algebraic structure of (S, +, ×). Similar to a ring, but without the condition that each element must have an inverse to the adding operation. The forms (S, +) and (S, ×) are semigroups that satisfy the distributive law of multiplication and addition. In matrix theory, there is a term known as the Kronecker product. This operation transforms two matrices into a larger matrix containing all possible products of the entries in the two matrices. This Kronecker product has several properties often used to solve the complex problems of linear algebra and its applications. The Kronecker product is related to the Lyapunov equation of a linear system. Based on previous research in the Lyapunov equation in conventional linear algebra, this paper will describe the characteristics of the Lyapunov equation in a semiring linear system in terms of the Kronecker product. [ABSTRACT FROM AUTHOR]

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are "almost idempotent" to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question. [ABSTRACT FROM AUTHOR]

Many uncertainties arise in real-world problems, making them impossible to solve using conventional approaches. Researchers all over the world have developed new mathematical theories like fuzzy set theory and rough set theory to better understand the uncertainties that occur in various fields. Soft set theory, which was recently introduced, offers a novel approach to real-world problem solving by removing the need to set the membership function. This is helpful in resolving a variety of issues, and much progress is being made these days. Recently, Jun introduced the concept of a hybrid structure, which blends the concepts of a fuzzy set as well as a soft set. In this paper, we define the hybrid k-sum and hybrid k-product of k-ideals of semiring and investigate their properties. We illustrate with an example that the hybrid sum and hybrid product of two k-ideals are not always hybrid ideals. We also describe semiring regularity constraints in terms of hybrid k-ideal structures. [ABSTRACT FROM AUTHOR]

Let M be a finitely generated free semimodule over a semiring S with identity having invariant basis number property with a basis α = {α1,..., αk}. The complement Γ ∗ ¯ (M) of the reduced non-zero component graph Γ ∗ (M) of M , is the simple undirected graph with V = M ∗ \ { ∑ i = 1 k c i α i : c i ≠ 0 ∀ i } as the vertex set and such that there is an edge between two distinct vertices a = ∑ i = 1 k a i α i and b = ∑ i = 1 k b i α i if and only if there exists no i such that both ai, bi are non-zero. In this paper, we show that the graph Γ ∗ ¯ (M) is connected and find its domination number, clique number and chromatic number. In the case of finite semirings, we determine the degree of each vertex, order, size, vertex connectivity and girth of Γ ∗ ¯ (M) . Also, we give a necessary and sufficient condition for Γ ∗ ¯ (M) to be Eulerian or Hamiltonian or planar. [ABSTRACT FROM AUTHOR]

The structural theory of semirings with additional conditions is considered. Multiplicatively idempotent semirings with the annihilator condition are studied. General properties of such semirings are considered. Examples are given. A criterion for the fulfillment of the annihilator condition in an arbitrary multiplicatively idempotent semiring with zero is proved (Proposition 6). In terms of annihilators, new abstract characterizations of semirings isomorphic to the direct product of a Boolean ring with unit and a Boolean lattice (Theorem 1) are obtained. The direct product of a Boolean ring and a distributive lattice with the annihilator condition is a multiplicatively idempotent semiring with the annihilator condition. Generally speaking, the opposite is not true (Theorem 2). An example of a general nature of a multiplicatively idempotent semiring with unit and with the annihilator condition, which is not isomorphic to the direct product of a Boolean ring and a distributive lattice, is constructed. In conclusion, a number of supplements are given. [ABSTRACT FROM AUTHOR]

Conditions are found on an operation on a finite set for its transform to be lacunary, that is, missing many expected terms. Often the condition is that the operation preserves a relation. General operations are split into odd and even lacunary parts, or more generally, into several lacunary parts given by a non-binary parity. This is applied to classical Fourier transforms as well as algebraic normal forms. With this theory, an explicit polynomial expansion is given for any operation in a preprimal algebra based on a finite elementary Abelian group. Convolution is defined, with a criterion given for it to commute with transforms. [ABSTRACT FROM AUTHOR]

This article defines almost symmetric and generalized symmetric semirings. These notions generalize symmetric semirings, as well as almost symmetric rings and pseudosymmetric rings. Isomorphic representations of these semirings by sections of the Lambek sheaf of semirings are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we define quasi-primary ideals in commutative semirings S with 1 ≠ 0 which is a generalization of primary ideals. A proper ideal I of a semiring S is said to be a quasi-primary ideal of S if ab ∈ √I implies a ∈ √I or b ∈ √I. We also introduce the concept of 2-absorbing quasi-primary ideal of a semiring S which is a generalization of quasi-primary ideal of S. A proper ideal I of a semiring S is said to be a 2-absorbing quasi-primary ideal if abc ∈ √I implies ab ∈ √I or bc ∈ √I or ac ∈ √I. Some basic results related to 2-absorbing quasi-primary ideal have also been given. [ABSTRACT FROM AUTHOR]

In this paper we prove that if R and S are Morita equivalent semirings via Morita context (R, S, P, Q, φ, σ), then there exists a one-to-one inclusion preserving correspondence between the set of all prime ((right) strongly prime, uniformly strongly prime) ideals of R and the set of all prime (resp. (right) strongly prime, uniformly strongly prime) subsemimodules of P. We also show that prime radicals, (right) strongly prime radicals, uniformly strongly prime radicals are preserved under Morita equivalence of semirings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a derivation of ordered semirings, study some properties of derivations of ordered semirings, derivations of fuzzy ideals and relations between derivation and homomorphism of ordered semirings. We prove that if μ is a fuzzy prime ideal and d is an onto derivation of idempotent ordered semiring M with identity then d(μ) and d-1(μ) are fuzzy prime ideals of M. [ABSTRACT FROM AUTHOR]