scholarly journals On some small varieties of distributive Ockham algebras

1984 ◽  
Vol 25 (2) ◽  
pp. 175-181 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Duality Theory ◽  
Basic Method ◽  
Distributive Lattices ◽  
Algebraic Proof ◽  
Subdirectly Irreducible ◽  
Topological Duality ◽  
Hasse Diagrams ◽  
Ockham Algebras ◽  
Subdirectly Irreducibles ◽  
Subdirectly Irreducible Algebras

J. Berman [2] initiated the study of a variety k of bounded distributive lattices endowed with a dual homomorphic operation paying particular attention to certain subvarieties km, n. Subsequently, A. Urquhart [8] named the algebras in k distributive Ockham algebras, and developed a duality theory, based on H. A. Priestley's order-topological duality for bounded distributive lattices [6], [7]. Amongst other things, Urquhart described the ordered spaces dual to the subdirectly irreducible algebras in Sif. This work was developed further still by M. S. Goldberg in his thesis and the paper [5]. Recently, T. S. Blyth and J. C. Varlet [3], in abstracting de Morgan and Stone algebras, studied a subvariety MS of the variety k1.1. The main result in [3]is that there are, up to isomorphism, nine subdirectly irreducible algebras in MS and their Hasse diagrams are exhibited. The methods employed in [3] are purely algebraic and can be generalized to show that, up to isomorphism, there are twenty subdirectly irreducible algebras in k1.1. In section 3 of this paper, we take a short cut to this result by utilizing the results of Urquhart and Goldberg. Our basic method is simple: the results of Goldberg [5] are applied to k1,1 to produce a certain eight-element algebra B1 in k1,1, whose lattice reduct is Boolean and whose subalgebras are, up to isomorphism, precisely the subdirectly irreducibles in k1.1. We then pick out of the list of twenty such algebras those belonging to the variety MS. In section 4, we sketch a purely algebraic proof along the lines followed by Blyth and Varlet in [3].

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.

scholarly journals On injectives in some varieties of Ockham algebras

1993 ◽  
Vol 35 (3) ◽  
pp. 345-351
Author(s):  
Teresa Almada
Keyword(s):  
Duality Theory ◽  
Distributive Lattices ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Topological Duality ◽  
Subdirectly Irreducible Algebra ◽  
Lattice Of Subvarieties ◽  
Ockham Algebras

The study of bounded distributive lattices endowed with an additional dual homomorphic operation began with a paper by J. Berman [3]. Subsequently these algebras were called distributive Ockham lattices and an order-topological duality theory for them was developed by A. Urquhart [12]. In [9], M. S. Goldberg extended this theory and described the injective algebras in the subvarieties of the variety O of distributive Ockham algebras which are generated by a single subdirectly irreducible algebra. The aim here is to investigate some elementary properties of injective algebras in join reducible members of the lattice of subvarieties of Kn,1 and to give a complete description of injectivealgebras in the subvarieties of the Ockham subvariety defined by the identity x Λ f2n(x) = x.

scholarly journals Distributive Ockham algebras: free algebras and injectivity

1981 ◽  
Vol 24 (2) ◽  
pp. 161-203 ◽  
Author(s):  
Moshe S. Goldberg
Keyword(s):  
Distributive Lattices ◽  
Free Algebras ◽  
Subdirectly Irreducible ◽  
Irreducible Algebra ◽  
Subdirectly Irreducible Algebra ◽  
Ockham Algebras ◽  
Subdirectly Irreducibles ◽  
Priestley's Duality

This paper centres around the variety 0 of distributive Ockham algebras, and those subvarieties of 0 which are generated by a single finite subdirectly irreducible algebra A. We use H.A. Priestley's duality for bounded distributive lattices throughout. First, intrinsic descriptions of the duals of certain finite subdirectly irreducibles are given; these are later used to determine projectives in the dual categories. Next, left adjoints to the forgetful functors from 0 and Var(A) into bounded distributive lattices are obtained, thereby allowing us to describe all free algebras and coproducts of arbitrary algebras. Finally, by applying the duality, we characterize injectivity in Var(A) for each finite subdirectly irreducible algebra A.

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.

The Lattice of Congruences on a Subdirectly Irreducible Weak Stone-Ockham Algebra

2012 ◽  
Vol 19 (03) ◽  
pp. 545-552
Author(s):  
Jie Fang
Keyword(s):  
Stone Algebra ◽  
Subdirectly Irreducible ◽  
Ockham Algebra ◽  
Ockham Algebras ◽  
Lattice Of Congruences ◽  
Subdirectly Irreducible Algebras

Weak Stone-Ockham algebras are those algebras (L; ∧, ∨, f, ⋆,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, ⋆,0,1) is a weak Stone algebra, and the unary operations f and ⋆ commute. In this paper, we give a complete description of the structure of the lattice of congruences on the subdirectly irreducible algebras.

Ockham Algebras with Balanced Demi-pseudocomplementation

2010 ◽  
Vol 17 (04) ◽  
pp. 595-610
Author(s):  
Jie Fang
Keyword(s):  
Subdirectly Irreducible ◽  
Ockham Algebra ◽  
Ockham Algebras ◽  
Subdirectly Irreducible Algebras

Let bdpO be the variety consisting of those algebras (L; ∧, ∨, f, *,0,1) of type 〈2,2,1,1,0,0〉, where (L; ∧, ∨, f,0,1) is an Ockham algebra, (L; ∧, ∨, *,0,1) is a demi-pseudocompleted lattice, and the unary operations f and * satisfy the conditions [f(x)]*=f2(x) and f(x*)=x**. Here we give a description of the structure of the subdirectly irreducible algebras in bdpO.

scholarly journals Varieties of distributive lattices with unary operations I

1997 ◽  
Vol 63 (2) ◽  
pp. 165-207 ◽  
Author(s):  
H. A. Priestley
Keyword(s):  
Duality Theory ◽  
Algebraic Logic ◽  
Distributive Lattices ◽  
Ordered Semigroup ◽  
Finitely Generated ◽  
Left Multiplication ◽  
Unified Study

AbstractA unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of the form (P; ∧, ∨, 0, 1, {fμ}), where each fμ is an endomorphism or dual endomorphism of (P; ∧, ∨, 0, 1), and the Priestly dual of this lattice is an ordered semigroup N whose elements act by left multiplication to give the maps dual to the operations fμ. Duality theory is fully developed within this framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for HSP () to be essentially the same, so that, inter alia, coproducts in HSP () are enriched D-coproducts.

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