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

Abstract: The concept of a $\Sigma$-closed subset was introduced earlier by the authors for an algebraic structure $\Cal A=(A,F,R)$ of type $\tau$ and a set $\Sigma$ of open formulas of the first order language $L(\tau)$. The set $C_\Sigma(\Cal A)$ of all $\Sigma$-closed subsets of $\Cal A$ forms a complete lattice whose properties were investigated in two previous papers by the authors. An algebraic structure $\Cal A$ is called $\Sigma$- hamiltonian, if every non-empty $\Sigma$-closed subset of $\Cal A$ is a class (block) of some congruence on $\Cal A$;$\Cal A$ is called $\Sigma$- regular, if $\theta=\Phi$ for every two $\theta$, $\Phi\in\operatorname{Con}\Cal A$ whenever they have a congruence class $B\in C_\Sigma(\Cal A)$ in common. This paper contains some results connected with $\Sigma$-regularity and $\Sigma$-hamiltonian property of algebraic structures.

Keywords: algebraic structure, closure system, $\Sigma$-closed subset, $\Sigma$-hamiltonian and $\Sigma$-regular algebraic structure, $\Sigma$-transferable congruence

Classification (MSC91): 08A05, 04A05

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