Expansions of semi-Heyting algebras. I: Discriminator varieties
This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. The paper has the following objectives:{\parindent=6mm \begin{itemize}\item[(1)] To prove the validity of the following conjecture: There exists a variety \textbf{V} of algebras which would provide a unifying framework to state and prove results which would be either common generalizations of strikingly similar results, or generalizations of other results. \item[(2)] To give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. \item[(3)] To investigate the variety \textbf{DblSH} of double semi-Heyting algebras and show that it too contains a rich supply of discriminator varieties. \item[(4)] To give applications of these results (to give bases for ``small'' varieties of \textbf{BDQDSH} -- blended dual quasi-De Morgan semi-Heyting algebras; \textbf{DQSSH} -- dually quasi-Stone semi-Heyting algebras; and \textbf{DblSH} -- double semi-Heyting algebras). \end{itemize}} The main results of the paper are the following: {\parindent=6mm\begin{itemize}\item[--] Definition of the new variety \textbf{DHMSH} of dually quasi-De Morgan semi-Heyting algebras, including the subvariety of \textbf{DQDSH} of dually quasi-De Morgan semi-Heyting algebras, and presentation of some arithmetical properties of these varieties. \item[--] Introduction of the blended \(\vee\)-De Morgan law and the variety \textbf{BDQDSH} and proving that the variety \textbf{DPCSH} of dually pseudocomplemented semi-Heyting algebras and the variety \textbf{DmsSH} of dually semi-Heyting algebras are subvarieties of \textbf{BDQDSH}. \item[--] Proving that congruences of \textbf{DHMSH} are determined by normal filters. \item[--] Characterization of directly indecomposables and simples in the variety \textbf{BDQDSH}. \item[--] Proving that every finite simple \textbf{BDQDSH} is functionally complete. \item[--] Giving a sufficient condition on a very general expansion of semi-Heyting algebras by a unary operation to be a discriminator variety. \item[--] Description of the variety \textbf{DQDBSH} of dually quasi-De Morgan Boolean semi-Heyting algebras as the variety generated by three 4-elements \textbf{D1}, \textbf{D2} and \textbf{D3}. \item[--] Introduction of the variety \textbf{RDQDSH} of regular dually quasi-De Morgan semi-Heyting algebras and proving that \textbf{RDQDSH} is a subvariety of \textbf{BDQDSH}. \item[--] Giving a base for the subvariety \textbf{DQDSHC3} of \textbf{DQDDSH} generated by all twenty 3-chains, as well as bases for each of the 3-chains. \item[--] Presentation of a base for the variety \textbf{DMHC} generated by De Morgan Heyting chains, as well as a base for each of its subvarieties generated by a finite chain. \item[--] Giving a base for the variety \textbf{DPCHC} generated by dually pseudocomplemented Heyting chains and a base for each of its subvarieties generated by a finite chain. \item[--] Introduction of the subvariety \textbf{DQSSH} (dually quasi-Stone semi-Heyting algebras) of \textbf{DHMSH}, which is not a subvariety of \textbf{BDQDSH}. \item[--] Characterization of simples in \textbf{DQSSH} and proving that the variety \textbf{DQSSH} is a discriminator variety. \item[--] Investigation of the variety \textbf{DPCHC} as a subvariety of \textbf{DQSSH}. \item[--] Presentation of a base for \textbf{DPCHC} relative to \textbf{DQSSH}. \item[--] Description of an expansion of semi-Heyting algebras by a unary operation, namely the variety \textbf{DblSH}, and proving that their congruences are the same as those of their \textbf{DPCSH}-reducts. \item[--] Characterization of the directly indecomposables and simples in the variety \textbf{DblSH}. \item[--] Introduction of several subvarieties of \textbf{DblSH}, including \textbf{DblSH\(_n\)}, and proving that \textbf{DblSH\(_n\)} for \(n\in \omega\), form an increasing sequence of discriminator varieties. \item[--] Giving a base for each 2-element algebra, as well as a base for the variety generated by the 100 3-element chains, viewed as \textbf{DblSH}. \end{itemize}} I consider that the reviewed article contains valuable results and it is a starting point for other studies on this subject.
scientific article; zbMATH DE number 6013478
scientific article