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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]

Here, we have considered two equivalence relations λ ℓ and λ r that are congruences on regular orthogroups but not on completely regular semigroups, in general. The congruence openings λ ℓ o and λ r o of λ ℓ and λ r , respectively, induce new subvarieties λ ℓ o , λ ℓ o , λ r o , λ r o , and for every ∈ ℒ (ℛ) ; and are such that = λ ℓ o ∨ λ r o and = λ ℓ o ∨ λ r o . The decompositions of an arbitrary completely regular semigroup which are presented here generalize several spined product decompositions for particular classes of completely regular semigroups already given by several authors [M. Petrich and N. R. Reilly, Completely Regular Semigroup (Wiley, New York, 1999)], and thus contributes to a unification of the theory. [ABSTRACT FROM AUTHOR]

We associate a semigroup B(S) to every semiring S with semilattice additive reduct, namely the semigroup of all k-bi-ideals of S; and such semirings S have been characterized by this associated semigroup B(S). A semiring S is k-regular if and only if B(S) is a regular semigroup. For the left k-Clifford semirings S, B(S) is a left normal band; and consequently, B(S) is a semilattice if S is a k-Clifford semiring. Also we show that the set Bm(S) of all minimal k-bi-ideals of S forms a rectangular band and Bm(S) is a bi-ideal of the semigroup B(S).

Let G be a finite group of order n. The strong power graph Ps(G) of G is the undirected graph whose vertices are the elements of G such that two distinct vertices a and b are adjacent if am1=bm2 for some positive integers m1, m2 < n. In this article we classify all groups G for which Ps(G) is a line graph. Spectrum and permanent of the Laplacian matrix of the strong power graph Ps(G) are found for any finite group G.

In this paper, we introduce the bit graph Γ(B2n) of Boolean vectors in Bn2. The graph is both Eulerian and Hamiltonian for every n ≥3. Besides basic character-izations, we have determined the automorphism group and the clique number of the graph Γ(B2n). [ABSTRACT FROM AUTHOR]

Bandelt and Petrich [Subdirect products of rings and distributive lattices, Proc. Edinburgh Math. Soc. (2)25(2) (1982) 155–171] characterized a class of additive inverse semirings which are subdirect products of a distributive lattice and a ring. The aim of this paper is to characterize a class of additively regular semirings which are subdirect products of an idempotent semiring and a b -lattice of skew-rings. [ABSTRACT FROM AUTHOR]

Here, we continue to characterize a recently introduced notion, le-modules R M over a commutative ring R with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga.158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec (M) of all prime submodule elements of M. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec (M) is connected if and only if R/Ann(M) contains no idempotents other than 0 ¯ and 1 ¯. Open sets in the Zariski topology for the quotient ring R/Ann(M) induces a base of quasi-compact open sets for the Zariski topology on Spec (M). Every irreducible closed subset of Spec (M) has a generic point. Besides, we prove a number of different equivalent characterizations for Spec (M) to be spectral. [ABSTRACT FROM AUTHOR]

Here, we characterize the linear operators that preserve factor rank of matrices over additively-idempotent multiplicatively-cancellative semirings. The main results in this paper generalize the corresponding results on the two element Boolean algebra [L. B. Beasley and N. J. Pullman, Boolean-rank-preserving opeartors and Boolean-rank-1 spaces, Linear Algebra Appl.59 (1984) 55–77] and on the max algebra [R. B. Bapat, S. Pati and S.-Z. Song, Rank preservers of matrices over max algebra, Linear Multilinear Algebra48(2) (2000) 149–164]; and hold on max-plus algebra and some other tropical semirings. [ABSTRACT FROM AUTHOR]

In this paper, nil extensions of simple regular ordered semigroups, left simple and right regular ordered semigroups, etc. have been characterized. Also, we describe the ordered semigroups which are complete semilattices of nil extensions of left simple and right regular ordered semigroups, left group like ordered semigroups, etc. [ABSTRACT FROM AUTHOR]

The author [S. M. Yusuf, Inverse semimodules, J. Natur. Sci. Math. 6 (1966) 111–117.] introduced inverse semimodules. In this paper, we introduce Clifford semimodules. The Clifford semimodules have been characterized here as a strong R -semilattice of R -modules. The congruences on inverse semimodules, specially, the R -module congruences, have been characterized here completely. [ABSTRACT FROM AUTHOR]

In this article, we introduce and characterize linear dependence graph Γ(V) of a finite dimensional vector space V over a finite field of q elements. Two vector spaces U and V are isomorphic if and only if their linear dependence graphs Γ(U) and Γ(V) are isomorphic. The linear dependence graph Γ(V) is Eulerian if and only if q is odd. Highly symmetric nature of Γ(V) is reflected in its automorphism group ..., where m = .... Besides these basic characterizations of Γ(V), the main contribution of this article is to find eigen values of adjacency matrix, Laplacian matrix and distance matrix of this graph. [ABSTRACT FROM AUTHOR]

In this paper we give a purely combinatorial proof of an ordered-partition expansion of determinants inspired by multivariate finite operator calculus. This argument also includes a combinatorial proof of an interesting identity about Stirling numbers of the second kind. Also, we give a topological proof of Ryser's formula for permanents. [ABSTRACT FROM AUTHOR]

We associate a semigroup B(S) to every semiring S with semilattice additive reduct, namely the semigroup of all k-bi-ideals of S; and such semirings S have been characterized by this associated semigroup B(S). A semiring S is k-regular if and only if B(S) is a regular semigroup. For the left k-Clifford semirings S, B(S) is a left normal band; and consequently, B(S) is a semilattice if S is a k-Clifford semiring. Also we show that the set Bm(S) of all minimal k-bi-ideals of S forms a rectangular band and Bm(S) is a bi-ideal of the semigroup B(S). [ABSTRACT FROM AUTHOR]

For a finite groupGwith a normal subgroupH, the normal subgroupHbased power graph ofG, denoted byΓH(G), is a graph withV(ΓH(G)) = (G∖H)⋃{e} and two distinct verticesaandbare adjacent ifaH = bmHorbH = anHfor somem,n∈ℕ. In this article, we obtain some basic characterizations of the normal subgroup based power graphs of groups and investigate relationship of the graphsΓH(G) and. Here it is proved that. Also we show that the graphΓH(G) is a Cayley graph of some group if and only if it is complete; andΓH(G) is planar if and only if |H| = 2 or 3 and. Besides, we prove thatΓH(G) is Eulerian if and only if |G|≡|H|(mod 2). [ABSTRACT FROM AUTHOR]