Subdirectly Irreducible Algebras in One Class of Algebras with One Operator and the Main Near-Unanimity Operation

2021 ◽  
Vol 42 (1) ◽  
pp. 206-216
Author(s):  
V. L. Usol’tsev
Keyword(s):  
Subdirectly Irreducible ◽  
Near Unanimity ◽  
Subdirectly Irreducible Algebras

scholarly journals The subdirect decomposition theorem for classes of structures closed under direct limits

1980 ◽  
Vol 30 (2) ◽  
pp. 171-179 ◽  
Author(s):  
Xavier Caicedo
Keyword(s):  
Subdirect Product ◽  
Decomposition Theorem ◽  
Subdirectly Irreducible ◽  
Subdirect Decomposition ◽  
Direct Limits ◽  
Subdirectly Irreducible Algebras

AbstractBy a theorem of G. Birkhoff, every algebra in an equationally defined class of algebras K is a subdirect product of subdirectly irreducible algebras of K. In this paper we show that this result is true for any class of structures. not necessarily algebraic, closed under isomorphisms and direct limits. Quasivarieties in the sense of Malcev are examples of such classes of structures. This includes Birkhoffs result as a particular case.

scholarly journals Injectives in some small varieties of ockham algebras

1984 ◽  
Vol 25 (2) ◽  
pp. 183-191 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Duality Theory ◽  
Free Algebras ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Double Negation ◽  
Topological Duality ◽  
De Morgan Algebras ◽  
Ockham Algebras ◽  
The One ◽  
Subdirectly Irreducible Algebras

The study of bounded distributive lattices endowed with an additional dual homomorphic operation began with a paper by J. Berman [3]. On the one hand, this class of algebras simultaneously abstracts de Morgan algebras and Stone algebras while, on the other hand, it has relevance to propositional logics lacking both the paradoxes of material implication and the law of double negation. Subsequently, these algebras were baptized distributive Ockham lattices and an order-topological duality theory for them was developed by A. Urquhart [13]. In an elegant paper [9], M. S. Goldberg extended this theory and, amongst other things, described the free algebras and the injective algebras in those subvarieties of the variety 0 of distributive Ockham algebras which are generated by a single finite subdirectly irreducible algebra. Recently, T. S. Blyth and J. C. Varlet [4] explicitly described the subdirectly irreducible algebras in a small subvariety MS of 0 while in [2] the order-topological results of Goldberg were applied to accomplish the same objective for a subvariety k1.1 of 0 which properly contains MS. The aim, here, is to describe explicitly the injective algebras in each of the subvarieties of the variety MS. The first step is to draw the Hasse diagram of the lattice AMS of subvarieties of MS. Next, the results of Goldberg are applied to describe the injectives in each of the join irreducible members of AMS. Finally, this information, in conjunction with universal algebraic results due to B. Davey and H. Werner [8], is applied to give an explicit description of the injectives in each of the join reducible members of AMS.

Maximal n-generated subdirect products

2016 ◽  
Vol 26 (01) ◽  
pp. 123-155
Author(s):  
Joel Berman
Keyword(s):  
Upper Bound ◽  
Finite Algebra ◽  
Equivalence Relations ◽  
Subdirectly Irreducible ◽  
Finite Algebras ◽  
Locally Finite Variety ◽  
Congruence Relations ◽  
Finite Set ◽  
Basic Parameters ◽  
Subdirectly Irreducible Algebras

For [Formula: see text] a positive integer and [Formula: see text] a finite set of finite algebras, let [Formula: see text] denote the largest [Formula: see text]-generated subdirect product whose subdirect factors are algebras in [Formula: see text]. When [Formula: see text] is the set of all [Formula: see text]-generated subdirectly irreducible algebras in a locally finite variety [Formula: see text], then [Formula: see text] is the free algebra [Formula: see text] on [Formula: see text] free generators for [Formula: see text]. For a finite algebra [Formula: see text] the algebra [Formula: see text] is the largest [Formula: see text]-generated subdirect power of [Formula: see text]. For every [Formula: see text] and finite [Formula: see text] we provide an upper bound on the cardinality of [Formula: see text]. This upper bound depends only on [Formula: see text] and these basic parameters: the cardinality of the automorphism group of [Formula: see text], the cardinalities of the subalgebras of [Formula: see text], and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of [Formula: see text]. Using this upper bound on [Formula: see text]-generated subdirect powers of [Formula: see text], as [Formula: see text] ranges over the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text], we obtain an upper bound on [Formula: see text]. And if all the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text] have congruence lattices that are chains, then we characterize in several ways those [Formula: see text] for which this upper bound is obtained.

scholarly journals Simple and subdirectly irreducibles bounded distributive lattices with unary operators

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Sergio Arturo Celani
Keyword(s):  
Distributive Lattices ◽  
Heyting Algebras ◽  
Subdirectly Irreducible ◽  
Modal Algebras ◽  
Heyting Algebras With Operators ◽  
Subdirectly Irreducibles ◽  
Subdirectly Irreducible Algebras

We characterize the simple and subdirectly irreducible distributive algebras in some varieties of distributive lattices with unary operators, including topological and monadic positive modal algebras. Finally, for some varieties of Heyting algebras with operators we apply these results to determine the simple and subdirectly irreducible algebras.

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.

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