An example concerning definable principal congruences

1977 ◽  
Vol 7 (1) ◽  
pp. 403-404 ◽  
Author(s):  
Stanley Burris
Keyword(s):  
Definable Principal Congruences

scholarly journals Varieties generated by finite BCK-algebras

1980 ◽  
Vol 22 (3) ◽  
pp. 411-430 ◽  
Author(s):  
William H. Cornish
Keyword(s):  
Positive Integer ◽  
Finite Algebras ◽  
Equationally Definable Principal Congruences ◽  
Definable Principal Congruences

Iséki's BCK-algebras form a quasivariety of groupoids and a finite BCK-algebra must satisfy the identity (En): xyn = xyn+1, for a suitable positive integer n. The class of BCK-algebras which satisfy (En) is a variety which has strongly equationally definable principal congruences, congruence-3-distributivity, and congruence-3-permutability. Thus, a finite BCK-algebra generates a 3-based variety of BCK-algebras. The variety of bounded commutative BCK-algebras which satisfy (En) is generated by n finite algebras, each of which is semiprimal.

scholarly journals Double MSn-algebras and double Kn.m-algebras

1993 ◽  
Vol 35 (2) ◽  
pp. 189-201 ◽  
Author(s):  
M. Sequeira
Keyword(s):  
Subdirectly Irreducible ◽  
Topological Duality ◽  
Equationally Definable Principal Congruences ◽  
Ockham Algebras ◽  
Definable Principal Congruences ◽  
Subdirectly Irreducible Algebras

AbstractThe variety O2 of all algebras (L; ∧, ∨, f, g, 0, 1) of type (2, 2, 1, 1, 0, 0) such that (L; ∧, ∨, f, 0, 1) and (L; ∧, ∨, g, 0, 1) are Ockham algebras is introduced, and, for n, m εℕ, its subvarieties DMSn, of double MSn-algebras, and DKn,m, of double Kn,m-algebras, are considered. It is shown that DKn,m has equationally definable principal congruences: a description of principal congruences on double Kn,m-algebras is given and simplified for double MSn-algebras. A topological duality for O2-algebras is developed and used to determine the subdirectly irreducible algebras in DKn,m and in DMSn. Finally, MSn-algebras which are reduct of a (unique) double MSn-algebra are characterized.

scholarly journals Definable principal congruences and solvability

2009 ◽  
Vol 157 (1) ◽  
pp. 30-49
Author(s):  
Paweł M. Idziak ◽  
Keith A. Kearnes ◽  
Emil W. Kiss ◽  
Matthew A. Valeriote
Keyword(s):  
Definable Principal Congruences

scholarly journals Self-Rectangulating Varieties of Type 5

1997 ◽  
Vol 07 (04) ◽  
pp. 511-540 ◽  
Author(s):  
Keith A. Kearnes ◽  
Ágnes Szendrei
Keyword(s):  
Extension Property ◽  
Congruence Extension Property ◽  
Finite Variety ◽  
Finitely Generated ◽  
Locally Finite ◽  
Term Operation ◽  
Locally Finite Variety ◽  
Definable Principal Congruences

We show that a locally finite variety which omits abelian types is self-rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type-set {5}. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the congruence extension property.

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