Journal of Algebraic Structures and Their Applications
Volume:9 Issue: 2, Summer-Autumn 2022

In this paper, we combine two binary operations $Gamma$-Extension and element splitting under special conditions, to extend binary matroids. For a given binary matroid $M$, we call a matroid obtained in this way a $Lambda$-Extension of $M$. We note some attractive properties of this matroid operation, particularly constructing a chordal matroid from a chordal binary matroid.

The notions of $xi$-inside GE-derivation and $xi$-outside GE-derivation on a GE-algebra are introduced and its properties are investigated. Conditions for a self-map on GE-algebra to be a $xi$-inside GE-derivation and a $xi$-outside GE-derivation are provided. The $xi$-inside GE-derivation or the $xi$-outside GE-derivation $varrho$ are used to form two sets $X_{(varrho = xi)}$ and ${rm ker}(varrho)$, and GE-subalgebra and GE-filter are studied for these two sets.

This paper is about the extension of some classical separation axioms Hausdorffness, regularity and complete regularity to topoframes. We show that they agree with those in frames except perhaps for complete regularity. The interesting results are about complete regularity, in particular when and how these differ from the frame results. These together with the results about B-filters are the focus of the paper.

In [On a generalization of weak flatness property, Asian-European Journal of Mathematics, 14(1) (2021)] we introduce a generalization of weak flatness property, called $(WF)'$, and showed that a monoid $S$ is absolutely $(WF)'$ if and only if $S$ is regular and satisfies Conditions $(R_{(WF)'})$ and $(L_{(WF)'})$. In this paper we continue the characterization of monoids by this property of their (finitely generated, (mono)cyclic, Rees factor) right acts. Also we give a classification of monoids for which $(WF)'$ property of their (finitely generated, (mono)cyclic, Rees factor) right acts imply other properties and vise versa. The aim of this paper is to show that the class of absolutely $(WF)'$ monoids and absolutely (weakly) flat monids are coincide.

In this paper, the notion of an $M$-function and cut function on a set are introduced and investigated several properties. We use algebraic properties to introduce an algorithm which show that every finite $MV$-algebras and Fibonacci sequences determines a block-code and presented some connections between Fibonacci sequences, $MV$-algebras and binary block-codes. Furthermore, an $MV$-algebra arising from block-codes is established.

Let $C_c(X)$ be the functionally countable subalgebra of $C(X)$. Essential $CP$-spaces are introduced and investigated algebraically and topologically. It is shown that if $X$ is a proper essential $CP$-space, then $mC_c(X)$ is compact if and only if ${ eta }$ is a $G_delta$, where $eta$ is the only non $CP$-point of $X$ and $mC_c(X)$ is the space of minimal prime ideals of $C_c(X)$ which are not maximal. Quasi $F_c$-spaces, $c$-basically disconnect spaces, almost $CP$-spaces and almost essential $CP$-spaces are introduced and studied via essential $CP$-spaces. Finally, $C_c(X)$ as a $CSV$-ring where $X$ is an essential $CP$-space is investigated.

In this paper, we prove that projective special unitary groups $U_3 (3^n)$, where $ 3^{2n}-3^{n}+1$ is a prime number and $3^nequivpm2(mod 5)$, can be uniquely determined by the order of group and the number of elements with the same order.

In this paper, we introduce a proper generalization of that of Hopfian modules, called $gamma$-Hopfian modules. A right $R$-module $M$ is said to be $gamma$-Hopfian, if any surjective endomorphism of $M$ has a $gamma$-small kernel. Some basic characterizations of $gamma$-Hopfian modules are proved. We prove that a ring $R$ is semisimple cosingular if and only if every $R$-module is $gamma$-Hopfian. Further, we prove that the $gamma$-Hopfian property is preserved under Morita equivalences. Some other properties of $gamma$-Hopfian modules are also obtained with examples.

In this paper, we define a new notion of conjugacy in semigroups that reduces to the n-notion of conjugacy in an inverse semigroup. We compare our new notion with the existing notions. We characterize the notion in partial injective and in full injective transformations and determine the conjugacy classes in these semigroups.

In this paper, we introduce the concepts of $J$-prime ideals and $M_{J}$-ideals in posets, and obtain some of their interesting characterizations in posets. Furthermore, we discuss the properties of $J$-ideals that are analogous to $J$-prime ideals and $M_J$-ideals in posets. Finally, we establish a set of equivalent conditions for an ideal in a poset $mathcal{P}$ containing an ideal $J$ is an $J$-ideal, and for a semi-prime ideal $J$ to be an $M_{J}$-ideal of $mathcal{P}$.

Some properties of $sigma$-ideals of distributive lattices are studied. The classes of Boolean algebras, generalized Stone lattices, relatively complemented lattices are characterized with the help of $sigma$-ideals and maximal ideals. Some significant properties of prime $sigma$-ideals are studied with the help of a congruence.

Let $R$ be a commutative ring with identity $1 neq 0$ which admits at least two maximal ideals. In this article, we have studied simple, undirected graph $(operatorname{INC}(R))^{c}$ whose vertex set is the set of all proper ideals which are not contained in $J(R)$ and two distinct vertices $I_{1}$ and $I_{2}$ are joined by an edge in $(operatorname{INC}(R))^{c}$ if and only if $I_{1} subseteq I_{2}$ or $I_{2} subseteq I_{1}$. In this article, we have studied some interesting properties of $(operatorname{INC}(R))^{c}$.