## Locally coherent algebras.(English)Zbl 0958.08003

Let $${\mathbb A}$$ be an algebra with a constant unary term function $$0$$ and $${\mathbb V}$$ a variety with a constant unary term $$0$$. $${\mathbb A}$$ is called locally coherent if every subalgebra of $${\mathbb A}$$ which contains a class of some $$\Theta\in\text{ Con}{\mathbb A}$$ also contains $$\Theta$$. $${\mathbb A}$$ is called locally regular if every single class of every $$\Theta\in\text{Con}{\mathbb A}$$ determines $$\Theta$$. $${\mathbb A}$$ is said to have $$\text{LCUT}$$ if for every subalgebra $$(B,F)$$ of $${\mathbb A}$$, every $$b\in B$$, every $$\Theta\in\text{Con}{\mathbb A}$$ with $$[b]\Theta\subseteq B$$, every positive integer $$n$$ and every $$n$$-ary polynomial function $$p$$ over $${\mathbb A}$$ with $$p(b,\dots,b)=0$$, there holds $$p([b]\Theta,\dots,[b]\Theta)\subseteq B$$. $${\mathbb A}$$ is called permutable at $$0$$ if $$(\Theta\circ\Phi)=(\Phi\circ\Theta)$$ for all $$\Theta,\Phi\in\text{ Con}{\mathbb A}$$. $${\mathbb V}$$ is called locally coherent, locally regular or permutable at $$0$$ or it is said to have $$\text{LCUT}$$ if every of its members has the corresponding property. Locally coherent varieties are characterized by a Mal’tsev condition. Further it is proved that if $${\mathbb V}$$ is locally coherent then it both is locally regular and has $$\text{LCUT}$$, and that the converse holds if $${\mathbb V}$$ is permutable at $$0$$.

### MSC:

 08B05 Equational logic, Mal’tsev conditions 08A30 Subalgebras, congruence relations
Full Text:

### References:

  Chajda I.: Coherence, regularity and permutability of congruences. Algebra Universalis 17 (1983), 170-173. · Zbl 0537.08006  Chajda I.: Weak coherence of congruences. Czechoslovak Mathem. Journal 41 (1991), 149-154. · Zbl 0796.08003  Chajda I.: Locally regular varieties. Acta Sci. Math. (Szeged), 64 (1998), 431-435. · Zbl 0913.08006  Chajda I., Duda J.: Finitely generated relations and their applications to permutable and n-permutable relations. CMUC 23 (1982), 41-54. · Zbl 0497.08002  Chajda I., Eigenthaler G.: A remark on congruence kernels in complemented lattices and pseudocomplemented semilattices. Contributions to General Algebra 11, (J. Heyn, Klagenfurt), 1999, 55-58. · Zbl 0940.06009  Duda J.: Coherence in varieties of algebras. Czechoslovak Mathem. Journal 39 (1989), 711-716. · Zbl 0704.08003  Geiger D.: Coherent algebras. Notices Amer. Math. Soc. 21 (1974), A-436.  Gumm H.-P., Ursini A.: Ideals in universal algebras. Algebra Universalis 19 (1984), 45-54. · Zbl 0547.08001  Taylor W.: Uniformity of congruences. Algebra Universalis 3 (1974), 342-360. · Zbl 0313.08001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.