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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]

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]

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]

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]

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]

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]

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]

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]

We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally, we illustrate that there exist non-surjective linear transformations that preserve ρ or preserve Kleene stars of matrices or commute with the Kleene star operator. [ABSTRACT FROM AUTHOR]

We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a 3-element commutative example, which we show also has NP-hard membership for its variety. As well as being the only nonfinite axiomatisable ai-semiring on 3-elements, we are able to show that its nonfinite basis property infects many related semirings, including the natural ai-semiring structure on the semigroup B 2 1. We also extend previous group-theory based examples significantly, by showing that any finite additively idempotent semiring with a nonabelian nilpotent subgroup is not finitely axiomatisable for its identities. [ABSTRACT FROM AUTHOR]

The semigroup B0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_0$$\end{document} is the only 4-element subsemigroup of the 5-element Brandt semigroup B2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_2$$\end{document}, up to isomorphism. Being an inverse semigroup, the semigroup B2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_2$$\end{document} can naturally be considered an additively idempotent semiring and B0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_0$$\end{document} is its subsemiring. We show that the semiring B0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_0$$\end{document} has a finite basis of identities. [ABSTRACT FROM AUTHOR]

Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalized Choquet integral is defined. As special cases, pseudo-Choquet Stieltjes integrals, pseudo-fuzzy Stieltjes integrals, g -Choquet integrals, pseudo-(N)fuzzy integrals and pseudo-(S)fuzzy integrals are obtained, and various kinds of properties and convergence theorems are shown, meanwhile Markov, Jensen, Minkowski and Hölder inequalities are proved. In the end, the generalized discrete Choquet integral is discussed. The obtained results for the generalized Choquet integral cover some previous results on different types of nonadditive integrals. [ABSTRACT FROM AUTHOR]

The study of multiplicatively idempotent semirings with additional conditions is continued. It is proved that every multiplicatively idempotent semiring with ideal congruences is isomorphic to the direct product of a Boolean ring and a generalized Boolean lattice. Thus, a new abstract characterization is obtained for the direct products of Boolean rings and generalized Boolean lattices. Examples are given. [ABSTRACT FROM AUTHOR]

Many of the new fuzzy structures with complete M V -algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called A M V -algebras. The category of complete A M V -algebras is isomorphic to the category of special pairs (R , R ∗) of complete commutative semirings and the corresponding fuzzy sets are called (R , R ∗) -fuzzy sets. We use this theory to define (R , R ∗) -fuzzy relations, lower and upper approximations of (R , R ∗) -fuzzy sets by (R , R ∗) -relations, and rough (R , R ∗) -fuzzy sets, and we show that these notions can be universally applied to any fuzzy type structure that is transformable to (R , R ∗) -fuzzy sets. As an example, we also show how this general theory can be used to determine the upper and lower approximations of a color segment corresponding to a particular color. [ABSTRACT FROM AUTHOR]

A certain sets of graphs form semirings under some algebraic graph operations. Some properties of such semirings and the accompanying operations are dis- cussed in this paper; namely, conditions for regularity of semirings and determining the graph chromatic numbers are principal highlights. [ABSTRACT FROM AUTHOR]

We consider weighted tree automata over strong bimonoids (for short: wta). A wta A has the finite-image property if its recognized weighted tree language 〚 A 〛 has finite image; moreover, A has the preimage property if the preimage under 〚 A 〛 of each element of the underlying strong bimonoid is a recognizable tree language. For each wta A over a past-finite monotonic strong bimonoid we prove the following results. In terms of A 's structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta A it is decidable whether A has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that A has the preimage property. All our results also hold for weighted string automata. [ABSTRACT FROM AUTHOR]

A submodule W of V is summand absorbing, if x + y ∈ W implies x ∈ W , y ∈ W for any x , y ∈ V. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules. [ABSTRACT FROM AUTHOR]

In this note we show that pseudo-analysis tools can be effective in obtaining results in a distorted probability framework. More precisely, we introduce the notion of pseudo-independence and that of pseudo-moment generating function, the latter representing a generalization of the pseudo-Laplace transform, and both aiming at extending the corresponding notions in the usual probabilistic context. We show that these concepts and their properties, and more in general pseudo-analysis, are particularly useful to provide characterization results for a class of bivariate random vectors that we call "pseudo-Schur constant" family which represents an extension of the Schur-constant class. [ABSTRACT FROM AUTHOR]

Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. Conversely, we show that the systems arising in this way, called hypersystems , are characterized by certain elimination axioms. Systems are preserved under standard algebraic constructions; for instance matrices and polynomials over hypersystems are systems, but not hypersystems. We illustrate these results by discussing several examples of systems and hyperfields, and constructions like matroids over systems. [ABSTRACT FROM AUTHOR]

It is well-known that the lattice of all submodules of a module is modular. However, this is not the case for the lattice of subsemimodules of a semimodule. We show examples and describe these lattices for a given semimodule. We study closed and splitting subsemimodules and submodules of a given semimodule or module M , respectively. We derive a sufficient condition under which the lattice L c (M) of closed subsemimodules is a homomorphic image of the lattice L (M) of all subsemimodules. We describe the ordered set of splitting submodules of a module and show a natural bijective correspondence between this poset and the poset of all projections of this module. We show that this poset is orthomodular. This result extends the case known for the poset of closed subspaces of a Hilbert space which is used in the logic of quantum mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we deal with (m , n) -hyperideals of ordered semihyperrings. First, we provide some properties about these (m , n) -hyperideals. Next, we provide nontrivial examples of (m , n) -hyperideals that are not left/right hyperideals. Finally, we apply our results to ordered semirings and provide nontrivial examples on (m , n) -ideals that are not left/right ideals. [ABSTRACT FROM AUTHOR]

Structure properties of multiplicatively idempotent semirings are considered. Basic results concerning completely prime ideals in multiplicatively idempotent semirings are obtained. The main theorem describes the commutative multiplicatively idempotent semirings with zero in which all completely prime ideals are maximal: up to isomorphism, such semirings are exhausted by direct products of a Boolean ring and a generalized Boolean lattice. Examples are given showing that the conditions of commutativity and the presence of zero are essential. [ABSTRACT FROM AUTHOR]

Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the prime k-spectrum of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings. [ABSTRACT FROM AUTHOR]

An R -module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, a submodule W of V is "summand absorbing" (SA), if, for all x , y ∈ V , x + y ∈ W ⇒ x ∈ W , y ∈ W. These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation. [ABSTRACT FROM AUTHOR]

In this paper, we present results on the homological classification of semirings for the last ten years and certain new results. [ABSTRACT FROM AUTHOR]

In the spirit of our earlier articles on q-! special functions, the purpose of this article is to present many new q-number systems, which are based on the q-addition, which was introduced in our previous articles and books. First, we repeat the concept biring, in order to prepare for the introduction of the q-integers, which extend the q-natural numbers from our previous book. We formally introduce a q-logarithm for the q-exponential function for later use. In order to find q-analogues of the corresponding formulas for the generating functions and q-trigonometric functions, we also introduce q-rational numbers. Then the so-called q-real numbers Rq, with a norm, a q-deformed real line, and with three inequalities, are defined. The purpose of the more general q-real numbers Rq is to allow the other q-addition too. The closely related JHC q-real numbers Rq have applications to several q-Euler integrals. This brings us to a vector version of the q-binomial theorem from a previous paper, which is associated with a special case of the q-Lauricella function. New q-trigonometric function formulas are given to show the application of this umbral calculus. Then, some equalities between q-trigonometric zeros and extreme values are proved. Finally, formulas and graphs for q-hyperbolic functions are shown. [ABSTRACT FROM AUTHOR]

Let R be a commutative semiring with 1 ≠ 0 , M an R-semimodule, and n ≥ 1 a positive integer. A proper subsemimodule P of M is called n-absorbing, if whenever r 1 , ... , r n ∈ R and x ∈ M together with r 1 ... r n x ∈ P , imply r 1 ... r n ∈ (P : M) or there exists 1 ≤ i ≤ n such that r 1 ... r i ^ ... r n x ∈ P . In this paper, we study the concept of n-absorbing subsemimodules that is a generalization of prime subsemimodules. Some basic properties of these subsemimodules and some useful examples of them are investigated. Also, we study the stability of n-absorbing subsemimodules with respect to some various constructions of semimodules. [ABSTRACT FROM AUTHOR]

Let R be a commutative semiring with nonzero identity and M an R -semimodule. In this paper, we introduce the concept of ϕ -primal subsemimodule of M that is a generalization of primal ideal of a commutative ring. Then we give some examples and properties of these subsemimodules. Also, some characterizations of ϕ -primal subsemimodules are presented. [ABSTRACT FROM AUTHOR]

We study the unitary Cayley graph of a matrix semiring. We find bounds for its diameter, clique number and independence number, and determine its girth. We also find the relationship between the diameter and the clique number of a unitary Cayley graph of a semiring S and a matrix semiring over S. [ABSTRACT FROM AUTHOR]