$\Sigma$-Hamiltonian and $\Sigma$-regular algebraic structures

Abstract

The concept of a $\SS$-closed subset was introduced in [1] for an algebraic structure $\A=(A,F,R)$ of type $\t$ and a set $\SS$ of open formulas of the first order language $L(\t)$. The set $C_\SS(\A)$ of all $\SS$-closed subsets of $\A$ forms a complete lattice whose properties were investigated in [1] and [2]. An algebraic structure $\A$ is called $\SS$- hamiltonian, if every non-empty $\SS$-closed subset of $\A$ is a class (block) of some congruence on $\A$; $\A$ is called $\SS$- regular, if $\0=\F$ for every two $\0$, $\F\in\Con\A$ whenever they have a congruence class $B\in C_\SS(\A)$ in common. This paper contains some results connected with $\SS$-regularity and $\SS$-hamiltonian property of algebraic structures.