On infinite subdirectly irreducible algebras in locally finite equational classes

1981 ◽  
Vol 13 (1) ◽  
pp. 393-394 ◽  
Author(s):  
Wiesław Dziobiak
Keyword(s):  
Subdirectly Irreducible ◽  
Locally Finite ◽  
Subdirectly Irreducible Algebras

Regular double p-algebras

2019 ◽  
Vol 69 (1) ◽  
pp. 15-34 ◽  
Author(s):  
M. E. Adams ◽  
Hanamantagouda P. Sankappanavar ◽  
Júlia Vaz de Carvalho
Keyword(s):  
Free Algebra ◽  
Explicit Description ◽  
Priestley Duality ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
Equational Basis ◽  
Proper Subvariety ◽  
Simple Algebras ◽  
Countably Infinite ◽  
Subdirectly Irreducible Algebras

Abstract In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

scholarly journals Simple surjective algebras having no proper subalgebras

1990 ◽  
Vol 48 (3) ◽  
pp. 434-454 ◽  
Author(s):  
Ágnes Szendrei
Keyword(s):  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Minimal Variety ◽  
Locally Finite ◽  
Subdirectly Irreducible Algebra ◽  
Matrix Power

AbstractWe prove that every finite, simple, surjective algebra having no proper subalgebras is either quasiprimal or affine or isomorphic to an algebra term equivalent to a matrix power of a unary permutational algebra. Consequently, it generates a minimal variety if and only if it is quasiprimal. We show also that a locally finite, minimal variety omitting type 1 is minimal as a quasivariety if and only if it has a unique subdirectly irreducible algebra.

SOME PROPERTIES OF FINITELY DECIDABLE VARIETIES

1992 ◽  
Vol 02 (01) ◽  
pp. 89-101 ◽  
Author(s):  
MATTHEW A. VALERIOTE ◽  
ROSS WILLARD
Keyword(s):  
Order Theory ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
First Order ◽  
Minimal Sets ◽  
First Order Theory ◽  
Finite Member ◽  
Transfer Principles

Let [Formula: see text] be a variety whose class of finite members has a decidable first-order theory. We prove that each finite member A of [Formula: see text] satisfies the (3, 1) and (3, 2) transfer principles, and that the minimal sets of prime quotients of type 2 or 3 in A must have empty tails. The first result has already been used by J. Jeong [9] in characterizing the finite subdirectly irreducible members of [Formula: see text] with nonabelian monolith. The second result implies that if [Formula: see text] is also locally finite and omits type 1, then [Formula: see text] is congruence modular.

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.

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