Abstract
This chapter presents varieties of algebras with principal compact blocks (PCB). PCB is a Mal’cev condition. An algebra A has PCB if for any a, b, and c∈A, there exist p,q∈A such that [a,b,c] = [p,q]. A variety V has PCB whenever each A∈V has PCB. An algebra A has principal compact congruences (PCC) if every compact congruence on A is principal. A variety V has PCC whenever each A∈V has PCC. P. Zlatos proved that PCC varieties form a Mal’cev class. Identities for PCC varieties with permutable congruences were derived by I. Chajda. An algebra A has principal compactly generated congruences (PCGC) if every compactly generated congruence on A is principal. A variety V has PCGC whenever each A∈V has PCGC. PCGC varieties form a Mal’cev class. Polynomial pair characterizing this class consists of ternary polynomials; however, a sequence of such polynomial pairs with increasing arity ≧ 3 can be obtained.
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