Expansions of semi-Heyting algebras. I: Discriminator varieties
type: Item
instance of: scholarly article
title: Expansions of semi-Heyting algebras. I: Discriminator varieties
zbMATH Open document ID: 1254.06006
published in: Studia Logica
publication date: 2012-03-09T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08B26
zbMATH DE Number: 6013478
zbMATH Keywords: semi-Heyting algebra
zbMATH Keywords: De Morgan semi-Heyting algebra
zbMATH Keywords: blended \(\vee\)-De Morgan law
zbMATH Keywords: double semi-Heyting algebra
zbMATH Keywords: congruence
zbMATH Keywords: normal filter
zbMATH Keywords: discriminator variety
author: Hanamantagouda P. Sankappanavar
reviewed by: Lavinia Ciungu
describes a project that uses: Prover9
describes a project that uses: Mace4
MaRDI profile type: Publication
full work available at URL: s11225-011-9322-6
OpenAlex ID: W2090789107
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DOI: 10.1007/S11225-011-9322-6
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