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Reports on Mathematical Logic

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No. 34

Willem J. BLOK and Silvia B. LA FALCE

Komori Identities in Algebraic Logic
A b s t r a c t.
A variety generated by a class ${\mathmbb K}$ of BCK-algebras consists of BCK-algebras
if and only if it satisfies a certain kind of identity, first discovered by
Komori. A similar phenomenon is shown to hold more generally in a certain class
of quasivarieties of logic that includes not only the class of BCK-algebras
but also such classes as the quasivariety of biresiduation algebras and
quasivarieties of algebras with an equivalence operation. We describe a set of
identities (which we call {\it Komori identities\/}), and show that the
variety generated by a class
${\mathmbb K}$ of algebras in one of the quasivarieties considered is contained in the
quasivariety if and only if it satisfies a Komori identity. We use the result
to establish (i) that the subvarieties of any of the quasivarieties
studied are congruence 3-permutable and (ii) that the varietal join of two
subvarieties of any of the quasivarieties studied is contained in the
quasivariety.

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