Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety V of groupoids is the smallest variety containing V and the variety S of semilattices. Its quasiregularization is the smallest quasivariety containing V and S. In an earlier paper the authors described the lattice of quasivarieties of cancellative c...

In this paper, we  study the Arens regularity properties of module actions. We investigate some properties of topological centers of module actions ${Z}^ell_{B^{**}}(A^{**})$ and  ${Z}^ell_{A^{**}}(B^{**})$ with some conclusions in group algebras.

The trichotomy between regular, semiregular, and strongly irregular boundary points for $p$-harmonic functions is obtained unbounded open sets in complete metric spaces with a doubling measure supporting $p$-Poincar\'e inequality, $1<p<\infty$. We show that these are local properties. also deduce several characterizations of semiregular points. In particular, characterized by means capacity, me...

Let A be a Banach algebra and X be a Banach A-bimodule. In this paper, we define a new product on $Aoplus X$ and generalize the module extension Banach algebras. We  obtain characterizations of Arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new Banach algebra.