Expansions of Semi-Heyting Algebras I: Discriminator Varieties

Studia Logica 98 (1-2):27-81 (2011)
This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH . We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH . Finally, we apply these results to give bases for “small” subvarieties of BDQDSH , DQSSH , and DblSH
Keywords Dually hemimorphic semi-Heyting algebra  Dually pseudocomplemented semi-Heyting algebra  De Morgan semi-Heyting algebra  Blended ∨-De Morgan law  Blended dually quasi-De Morgan semi-Heyting algebra  Blended dually quasi-Stone semi-Heyting algebra  Double semi-Heyting algebra  congruence  normal filter  discriminator variety  simple  directly indecomposable  subdirectly irreducible  equational base
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DOI 10.1007/s11225-011-9322-6
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