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Zbl 0537.08006
Chajda, Ivan
Coherence, regularity and permutability of congruences.
(English)
[J] Algebra Univers. 17, 170-173 (1983). ISSN 0002-5240; ISSN 1420-8911/e

A variety ${\cal V}$ of algebras is regular, if any two congruences on each algebra ${\frak A}\in {\cal V}$ coincide whenever they have a congruence class in common. It is coherent, if for any two algebras ${\frak A,B}$ of ${\cal V}$ the condition that ${\frak A}\subseteq {\frak B}$ and ${\frak A}$ contains a class of a congruence $\theta$ on ${\frak B}$ implies that ${\frak A}$ is a union of classes of $\theta$. A variety ${\cal V}$ is permutable, if $\theta\sb 1\cdot \theta\sb 2=\theta\sb 2\cdot \theta\sb 1$ for any two congruences on every algebra ${\frak A}\in {\cal V}.$ \par An algebra ${\frak A}$ has subalgebras closed under principal congruence classes (briefly ${\frak A}$ is CUT), if for every subalgebra ${\frak B}$ of ${\frak A}$, all $x\in {\frak A},\quad y\in {\frak B},\quad z\in {\frak B}$ and any algebraic functions $\phi$ over ${\frak A}$ the condition $[z]\sb{\theta}\subseteq {\frak B}$ and $\phi(z,...,z)=y$ implies $\phi([z]\sb{\theta})\in {\frak B},$ where $\theta =\theta(x,y)$ and $[z]\sb{\theta}$ denotes the class of $\theta$ containing z. \par The main result of the paper is the following theorem. For a variety ${\cal V}$ the following conditions are equivalent: (1) ${\cal V}$ is coherent, (2) ${\cal V}$ is CUT, regular and permutable, (3) there exist an $(n+1)$-ary polynomial h and ternary polynomials $t\sb i$ over ${\cal V}$ such that $t\sb i(x,x,z)=z,\quad i=1,...,n,\quad h(y,t\sb 1(x,y,z),...,t\sb n(x,y,z))=x.$
MSC 2000:
*08B05 Equational logic in varieties of algebras
08B10 Congruence modularity and generalizations in varieties of algebras
08A30 Subalgebras of general algebraic systems

Keywords: variety of algebras; subalgebras closed under principal congruence classes

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