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The Equationally-Defined Commutator in Quasivarieties Generated by Two-Element Algebras

Janusz Czelakowski

subject

CommutatorPure mathematicsQuasivariety010102 general mathematicsCharacterization (mathematics)01 natural sciencesCongruence (geometry)0103 physical sciences010307 mathematical physics0101 mathematicsAbelian groupElement (category theory)Finite setAssociative propertyMathematics

description

The notion of the equationally-defined commutator was introduced and thoroughly investigated in (Czelakowski, 2015). In this work the properties of the equationally-defined commutator in quasivarieties generated by two-element algebras are examined. It is proved: If a quasivariety Q is generated by a finite set of two-element algebras, then the equationally-defined commutator of Q is additive (Theorem 3.1) Moreover it satisfies the associativity law (Theorem 3.6). The second result is strengthened if the quasivariety is generated by a single two-element algebra 2: If Q = SP(2), then the equationally-defined commutator of Q universally validates one of the following laws: [x,y] = x^y or [x,y] = 0 (Theorem 3.9). In other words, any quasivariety generated by a single two-element algebra is either relatively congruence-distributive or Abelian. A syntactical characterization of all quasivarieties generated by finite sets of two-element algebras is also presented (Theorems 2.2–2.3).

yearjournalcountryeditionlanguage
2018-01-01
https://doi.org/10.1007/978-3-319-74772-9_5