On the structure of varieties with equationally definable principal congruences I

1982 ◽  
Vol 15 (1) ◽  
pp. 195-227 ◽  
Author(s):  
W. J. Blok ◽  
D. Pigozzi
Keyword(s):  
Equationally Definable Principal Congruences ◽  
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 Fregean subtractive varieties with definable congruence

2001 ◽  
Vol 71 (3) ◽  
pp. 353-366 ◽  
Author(s):  
Paolo Agliano
Keyword(s):  
Hilbert Algebras ◽  
Structure Theorems ◽  
Equationally Definable Principal Congruences ◽  
Subtractive Variety ◽  
Definable Principal Congruences

AbstractIn this paper we investigate subtractive varieties of algebras that are Fregean in order to get structure theorems about them. For instance it turns out that a subtractive variety is Fregean and has equationally definable principal congruences if and only if it is termwise equivalent to a variety of Hilbert algebras with compatible operations. Several examples are provided to illustrate the theory.

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