Compatible Idempotent Terms in Universal Algebra

Ivan Chajda; Antonio Ledda; Francesco Paoli

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2014)

  • Volume: 53, Issue: 2, page 35-51
  • ISSN: 0231-9721

Abstract

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In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term t ( x ) ) in every member of the given variety. Here, we try to give a unified account of this phenomenon. In particular, we investigate what happens when various congruence properties—like congruence distributivity, congruence permutability or congruence modularity—are not supposed to hold unrestrictedly in any 𝐀 ∈ 𝒱 , but only for congruence classes of values of the term operation t 𝐀 .

How to cite

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Chajda, Ivan, Ledda, Antonio, and Paoli, Francesco. "Compatible Idempotent Terms in Universal Algebra." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53.2 (2014): 35-51. <http://eudml.org/doc/262211>.

@article{Chajda2014,
abstract = {In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term $t(x)$) in every member of the given variety. Here, we try to give a unified account of this phenomenon. In particular, we investigate what happens when various congruence properties—like congruence distributivity, congruence permutability or congruence modularity—are not supposed to hold unrestrictedly in any $\mathbf \{A\}\in \mathcal \{V\}$, but only for congruence classes of values of the term operation $t^\{\mathbf \{A\}\}$.},
author = {Chajda, Ivan, Ledda, Antonio, Paoli, Francesco},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Congruence distributive variety; congruence modular variety; congruence permutable variety; idempotent endomorphism; congruence distributive varieties; congruence modular varieties; congruence permutable varieties; idempotent endomorphisms; idempotent term operations},
language = {eng},
number = {2},
pages = {35-51},
publisher = {Palacký University Olomouc},
title = {Compatible Idempotent Terms in Universal Algebra},
url = {http://eudml.org/doc/262211},
volume = {53},
year = {2014},
}

TY - JOUR
AU - Chajda, Ivan
AU - Ledda, Antonio
AU - Paoli, Francesco
TI - Compatible Idempotent Terms in Universal Algebra
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2014
PB - Palacký University Olomouc
VL - 53
IS - 2
SP - 35
EP - 51
AB - In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term $t(x)$) in every member of the given variety. Here, we try to give a unified account of this phenomenon. In particular, we investigate what happens when various congruence properties—like congruence distributivity, congruence permutability or congruence modularity—are not supposed to hold unrestrictedly in any $\mathbf {A}\in \mathcal {V}$, but only for congruence classes of values of the term operation $t^{\mathbf {A}}$.
LA - eng
KW - Congruence distributive variety; congruence modular variety; congruence permutable variety; idempotent endomorphism; congruence distributive varieties; congruence modular varieties; congruence permutable varieties; idempotent endomorphisms; idempotent term operations
UR - http://eudml.org/doc/262211
ER -

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