## Reports on Mathematical Logic

### 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.