Graduate studies at Western
Studia Logica 98 (1-2):27-81 (2011)
|Abstract||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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