Minimal generics from subvarieties of the clone extension of the variety of Boolean algebras

Jerzy P/lonka¹

¹ Institute of Mathematics Polish Academy of Sciences Kopernika 18 51-617 Wroc/law, Poland

Colloquium Mathematicum, Tome 112 (2008) no. 1, pp. 131-145.

Résumé

Let $\tau$ be a type of algebras without nullary fundamental operation symbols. We call an identity $\varphi\approx\psi$ of type $\tau$ clone compatible if $\varphi$ and $\psi$ are the same variable or the sets of fundamental operation symbols in $\varphi$ and $\psi$ are nonempty and identical. For a variety $\mathcal V$ of type $\tau$ we denote by ${\mathcal V}^{c}$ the variety of type $\tau$ defined by all clone compatible identities from $\mathop{\rm Id}\nolimits(\mathcal V)$. We call ${\mathcal V}^{c}$ the clone extension of $\mathcal V$. In this paper we describe algebras and minimal generics of all subvarieties of ${\mathcal{B}}^{c}$, where $\mathcal B$ is the variety of Boolean algebras.

Détail

DOI : 10.4064/cm112-1-6

Keywords: tau type algebras without nullary fundamental operation symbols call identity varphi approx psi type tau clone compatible varphi psi variable sets fundamental operation symbols varphi psi nonempty identical variety mathcal type tau denote mathcal variety type tau defined clone compatible identities mathop nolimits mathcal call mathcal clone extension mathcal paper describe algebras minimal generics subvarieties mathcal where mathcal variety boolean algebras