Journal of Linear and Topological Algebra
Volume:10 Issue: 3, Summer 2021

‎For a given topological space $(X‎, ‎~Im)$,‎ ‎there is a coarser topology on $X$ which is called the semi-regular topology on $X$ (generated by regularly open subsets) and it is denoted by $Im^{delta}$‎. ‎In this paper‎, ‎we study the continuity of the group operation and the inversion mapping ($varsigmalongmapstovarsigma^{-1}$) as regards the semi-regular topology $Im^{delta}$ (not necessarily with the given topology)‎. ‎Then we study the said mappings with the blend of the given topology $Im$ and the semi-regular topology $Im^{delta}$‎. ‎In the twilight of this note‎, ‎we pose some questions which are noteworthy‎.

In this work, we intend to introduce and study another algebraic structure of single-valued neutrosophic sets called neutrosophic field as a continuation of our investigations on neutrosophic algebraic structures. For this goal, we define the concept of neutrosophic fields and observe some of their basic characteristics and properties. Then we give the definition of a neutrosophic linear space over the proposed neutrosophic field and consider its fundamental properties.

‎Any notion of purity is normally defined in terms of‎ ‎solvability of some set of equations‎. ‎To study mathematical notions‎, ‎such as injectivity‎, ‎tensor products‎, ‎flatness‎, ‎one needs to have some categorical and‎ ‎algebraic information about the pair (${mathcal A}$,${mathcal M}$)‎, ‎for a category $mathcal A$‎ ‎and a class $mathcal M$ of monomorphisms in a category $mathcal A$‎. ‎In this paper we take $mathcal A$ to be the category {bf Act-S}‎ ‎of $S$-acts‎, ‎for a semigroup $S$‎, ‎and ${mathcal M}_{sp}$ to be‎ ‎the class of $C_I^{sp}$-pure monomorphisms and study some‎ ‎categorical and algebraic properties of this class concerning the closure operator $C_I^{sp}$‎.

‎Quasi module is a new algebraic structure‎, ‎based on module‎, ‎which is composed of a semigroup structure and a partial order accompanied with an external ring multiplication‎. ‎We proposed this structure in our paper cite{qmod} while we were studying the hyperspace $ com{M}{} $ consisting of all nonempty compact subsets of a topological module $ M $ over some topological ring $ R $‎. ‎Quasi module can be considered as a generalisation of module in some sense‎. ‎In the present paper we have defined topological quasi module and given some examples of it‎. ‎We have shown that the Cartesian product of arbitrary family of topological quasi modules is again a topological quasi module over some topological unitary ring‎. ‎Finally we have defined projective system of topological quasi modules and projective limit of this system‎. ‎We have proved various topological properties of the projective limit of a projective system‎.

‎For a given subspace as a solution space of a linear ODE‎, ‎we define a special linear parametric group action and prolong it to the jet bundle‎. ‎We determine these group parameters by moving frame method and prove that these group parameters are the first integrals of the given ODE‎. ‎These first integrals are used to construct the general form of operators which preserve given subspace invariant‎.

For a given positive integer n, the n th commutativity degree of a finite noncommutative semigroup S is defined to be the probability of choosing a pair (x, y) for x, y ∈ S such that x n and y commute in S. If for every elements x and y of an associative algebraic structure (S, .) there exists a positive integer r such that xy = y rx, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. In this paper, we study the n th commutativity degree of certain classes of quasi-commutative semigroups. We show that the n th commutativity degree of such structures is greater than 1 2 . Finally, we compute the n th commutativity degree of a finite class of non-quasi-commutative semigroups and we conclude that it is less than 1 2

‎Let ${A}$ be a Banach space and ${lambda}$ be a non-zero fixed element of ${A}^{ast}$(dual space of ${A}$) with non-zero kernel‎. ‎Defining algebra product in $A$ as $acdot b=lambda(a)b$ for $a,bin {A}$‎, ‎we show that ${A}$ is a $(2m-1)$-weakly amenable Banach algebra but not $2m$-weakly amenable for any $min{N}$‎. ‎Furthermore‎, ‎we show the converse of the statement [2,~Proposition,1.4.(ii)] ``for a non-unital Banach algebra $A$‎, ‎if $A$ is weakly amenable then $A^{#}$ is weakly amenable‎" ‎does not hold‎.