Medial modes, a natural generalization of normal bands, were investigated by Płonka. Rectangular algebras, a generalization of rectangular bands (diagonal modes) were investigated by Pöschel and Reichel. In this paper we show that each medial mode embeds as a subreduct into a semimodule over a certain ring, and that a similar theorem holds for each Lallement sum of cancellative modes over a medial mode. Similar results are obtained for rectangular algebras. The paper generalizes earlier results...

Let τ be a type of algebras without nullary fundamental operation symbols. We call an identity φ ≈ ψ of type τ clone compatible if φ and ψ are the same variable or the sets of fundamental operation symbols in φ and ψ are nonempty and identical. For a variety of type τ we denote by ${}^c$ the variety of type τ defined by all clone compatible identities from Id(). We call ${}^c$ the clone extension of . In this paper we describe algebras and minimal generics of all subvarieties of ${}^c$, where is the variety of...

Let $𝒜=(A,F,R)$ be an algebraic structure of type $\tau$ and $\Sigma$ a set of open formulas of the first order language $L\left(\tau \right)$. The set $C_{\Sigma }\left(𝒜\right)$ of all subsets of $A$ closed under $\Sigma$ forms the so called lattice of $\Sigma$-closed subsets of $𝒜$. We prove various sufficient conditions under which the lattice $C_{\Sigma }\left(𝒜\right)$ is modular or distributive.