By Janusz Czelakowski
This monograph introduces and explores the notions of a commutator equation and the equationally-defined commutator from the viewpoint of summary algebraic good judgment. An account of the commutator operation linked to equational deductive structures is gifted, with an emphasis put on logical features of the commutator for equational structures decided by means of quasivarieties of algebras. the writer discusses the overall homes of the equationally-defined commutator, a number of centralization relatives for relative congruences, the additivity and correspondence houses of the equationally-defined commutator and its habit in finitely generated quasivarieties.
Presenting new and unique examine no longer but thought of within the mathematical literature, The Equationally-Defined Commutator can be of curiosity to specialist algebraists and logicians, in addition to graduate scholars and different researchers attracted to difficulties of contemporary algebraic logic.
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Extra resources for The Equationally-Defined Commutator: A Study in Equational Logic and Algebra
Q/ˆ : C0 is thus the consequence operation determined by the set of all Q-valid equations and the rules of inference of Birkhoff’s logic B . ;// is a closed theory of C. e/ is also a theory of C0 . e/. 1. Let Q be a quasivariety of -algebras. and let e W Te ! Te be an epimorphism. Define T and the retraction k W Te ! T as above. e/. Proof. Let p; q 2 Te . ;/ is an invariant set of equations and g is an isomorphism from Te onto T. eq/ results from ep eq by renaming the variables occurring ep eq in a one-to-one way.
Y/ ( ) in the term algebra Te . Proof (of the lemma). T/. Claim 1. Y/T /. 3 More on Epimorphisms and the Equationally-Defined Commutator 49 Proof (of the claim). kW/T /; for all sets of equations Z; W in Eq. /. T/ and k is the identity map on the subalgebra T, the claim follows. t u Claim 2. Y/. Proof (of the claim). C0 . k. k/ CQ C. k/ CQ C. k/ CQ C. T/. t u From the claims the lemma follows. t u The proof of the theorem is concluded. 7. ) Let Q be a quasivariety of -algebras and e W Te ! Te an epimorphism.
10). 7). 22 holds for any relatively congruence-distributive quasivariety Q (without any restrictions imposed on X, Y and e). ) Let m and n be positive integers and let x D x1 ;: : :; xm , y D y1 ;: : :; ym , z D z1 ;: : :; zn , w D w1 ; : : : ; wn , and u D u1 ; : : : ; uk be sequences of pairwise distinct individual variables. The lengths of the strings x and y are equal, jxj D jyj D m and, similarly, jzj D jwj D n, juj D k. x1 ; : : : ; xm ; y1 ; : : : ; ym ; z1 ; : : : ; zn ; w1 ; : : : ; wn ; u1 ; : : : ; uk / be terms in Te built up with at most the variables x D x1 ; : : : ; xm , y D y1 ; : : : ; ym , z D z1 ; : : : ; zn , w D w1 ; : : : ; wn , and u D u1 ; : : : ; uk .