Journal of Algebraic Structures and Their Applications
Volume:5 Issue: 1, Winter - Spring 2018

In this paper, by considering the notion of Σ -hyperalgebras for an arbitrary signature Σ, we study the notions of regular and strongly regular relations on a Σ-hyperalgebra, A. We show that each regular relation which contains a strongly regular relation is a strongly regular relation. Then we concentrate on the connection between the fundamental relation of A and the set of complete parts of A

A mixed dominating set S of a graph G=(V,E) is a subset of vertices and edges like S⊆V∪E such that each element v∈(V∪E)∖S is adjacent or incident to at least one element in S. The mixed domination number γm(G) of a graph G is the minimum cardinality among all mixed dominating sets in G. The problem of finding γm(G) is known to be NP-complete. In this paper, we present an explicit polynomial-time algorithm using the parse tree to construct a mixed dominating set of size γm(G) where G is a generalized series-parallel graph.

Let R be a commutative ring with identity and M be a unital R-module. A proper submodule N of M with N:RM=p is said to be prime or p-prime (\p a prime ideal of R) if rx∈N for r∈R and x∈M implies that either x∈N or r∈p. In this paper we study a new equivalent conditions for a minimal prime submodules of an R-module to be a finite set, whenever R is a Noetherian ring. Also we introduce the concept of arithmetic rank of a submodule of a Noetherian module and we give an upper bound for it.

In this paper, some new properties of EQ -algebras are investigated. We introduce and study the notion of Boolean center of lattice ordered EQ-algebras with bottom element. We show that in a good ℓEQ-algebra E with bottom element the complement of an element is unique. Furthermore, Boolean elements of a good bounded lattice EQ-algebra are characterized. Finally, we obtain conditions under which Boolean center of an EQ-algebra E is the subalgebra of E

In this paper, we study the notions of \es~ and \ec~ modules and obtain some related results. For instance, we show that in a right self-injective ring R , all nonzero ideals of R are endo-semiprime as right (left) R-modules if and only if R is semiprime. Also, we prove that both being endo semiprime and being \ec~ are Morita invariant properties.

Given a binary matroid M and a subset T⊆E(M), Luis A. Goddyn posed a problem that the dual of the splitting of M, i.e., ((MT)∗) is not always equal to the splitting of the dual of M, ((M∗)T). This persuade us to ask if we can characterize those binary matroids for which (MT)∗=(M∗)T. Santosh B. Dhotre answered this question for a two-element subset T. In this paper, we generalize his result for any subset T⊆E(M) and exhibit a criterion for a binary matroid M and subsets T for which (MT)∗ and (M∗)T are the equal. We also show that there is no subset T⊆E(M) for which, the dual of element splitting of M, i.e., ((M′T)∗) equals to the element splitting of the dual of M, ((M∗)′T).