For an algebraic structure $=(A,F,R)$ or type $$ and a set $\Sigma$ of open formulas of the first order language $L\left(\right)$ we introduce the concept of $\Sigma$-closed subsets of $$. The set ${ℂ}_{\Sigma }\left(\right)$ of all $\Sigma$-closed subsets forms a complete lattice. Algebraic structures $$, $$ of type $$ are called $\Sigma$-isomorphic if ${ℂ}_{\Sigma }\left(\right)\cong {ℂ}_{\Sigma }\left(\right)$. Examples of such $\Sigma$-closed subsets are e.g. subalgebras of an algebra, ideals of a ring, ideals of a lattice, convex subsets of an ordered or quasiordered set etc. We study $\Sigma$-isomorphic algebraic structures in dependence...

For a class $K$ of structures and $A\in K$ let ${Con}^{*}\left(A\right)$ resp. ${Con}^K\left(A\right)$ denote the lattices of $*$-congruences resp. $K$-congruences of $A$, cf. Weaver [25]. Let ${Con}^{*}\left(K\right):=I\{{Con}^{*}\left(A\right):A\in K\}$ where $I$ is the operator of forming isomorphic copies, and ${Con}^r\left(K\right):=I\{{Con}^K\left(A\right):A\in K\}$. For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${Con}^{<}\left(A\right)$, and let ${Con}^{<}\left(K\right):=I\{{Con}^{<}\left(A\right):A\in K\}$ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^s$ and $P$, respectively. Let $\lambda$ be a lattice identity and let $\Sigma$ be a set of lattice identities....

Let $G\subset {ℝ}^m$ $(m\ge 2)$ be an open set with a compact boundary $B$ and let $\sigma \ge 0$ be a finite measure on $B$. Consider the space $L^1\left(\sigma \right)$ of all $\sigma$-integrable functions on $B$ and, for each $f\in L^1\left(\sigma \right)$, denote by $f\sigma$ the signed measure on $B$ arising by multiplying $\sigma$ by $f$ in the usual way. ${𝒩}_{\sigma }f$ denotes the weak normal derivative (w.r. to $G$) of the Newtonian (in case $m>2$) or the logarithmic (in case $n=2$) potential of $f\sigma$, correspondingly. Sharp geometric estimates are obtained for the essential norms of the operator ${𝒩}_{\sigma }-\alpha I$ (here $\alpha \in ℝ$...

Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences ${\rho }_{min},{\rho }_{max},{\rho }^{min}$ and ${\rho }^{max}$ on S and showed that $\rho \theta =[{\rho }_{min},{\rho }_{max}]$ and $\rho \kappa =[{\rho }^{min},{\rho }^{max}]$. Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if ${\rho }_{max}$ is a distributive lattice congruence and ${\rho }^{max}$ is a skew-ring congruence on S. If η (σ) is the...