Variety Of Distributive Lattices Research Articles

Finite axiomatizability of logics of distributive lattices with negation

Abstract This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite (i.e. that contain a finite number of finitary rule schemata). On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well.

The Ramsey and the Ordering Property for Classes of Lattices and Semilattices

The class of finite distributive lattices, as many other classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokic have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the variety of distributive lattices is not an exception, but an instance of a more general phenomenon. We show that for almost all nontrivial locally finite varieties of lattices no “reasonable” expansion of the finite members of the variety by linear orders gives rise to a Ramsey class. The responsibility for this lies not with the lattices as structures, but with the lack of algebraic morphisms: if we consider lattices as partially ordered sets (and thus switch from algebraic embeddings to embeddings of relational structures) we show that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property. It now comes as no surprise that the same is true for varieties of semilattices.

Idempotent n -permutable varieties

One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n-permutable for some n. In this paper, we prove two results: (1) for every n > 1 , there is a polynomial-time algorithm that, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is n-permutable and (2) a variety is n-permutable for some n if and only if it interprets an idempotent variety that is not interpretable in the variety of distributive lattices.

On distributive lattices and weakly Mal’tsev categories

We prove that a variety of lattices is weakly Mal’tsev if and only if it is a variety of distributive lattices.

A Method of Representing Rough Sets System Determined by Quasi Orders

In this paper, we study about the ordered structure of rough sets determined by a quasi order. A characterization theorem for rough sets of an approximation space (U, R) based on a quasi order R is given in Nagarajan and Umadevi (2010). Then using the characterization of rough sets determined by a quasi order, its rough sets system is represented by a new construction. This construction is generalized and abstracted into a new method of constructing Kleene based algebraic structures from dually isomorphic distributive lattices. Then by using different varieties of distributive lattices, we obtain various Kleene based algebraic structures. By this construction, we give various algebraic structures to the rough sets system determined by a quasi order R.

Subvarieties of the clone extension of the variety of distributive lattices

Let τ be a type of algebras without nullary fundamental operation symbols. An identity . φ ≈ ψ of type τ we call clone compatible if φ and ψ are the same variable or the sets of fundamental operation symbols in φ and ψ are non empty and identical. For a variety ν of type τ we denote by ν c the variety of type τ defined by all clone compatible identities from Id(ν). We call ν c the clone extension of ν. In this paper we describe identities, equational bases and minimal generics of all subvarieties of \( {\user1{{\mathcal{D}}}}^{c} \), where $$ {\user1{{\mathcal{D}}}} $$ is the variety of distributive lattices.

Simple and subdirectly irreducibles bounded distributive lattices with unary operators

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.

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics I

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes.

On n -clone extensions of algebras

We consider algebras of a given type \(\tau\) with a set F of fundamental operation symbols and without nullary operations. In this paper we generalize notions and results of [12]. An identity \(\varphi \approx \psi\) is called clone compatible if \(\varphi\) and \(\psi\) are the same variable or the sets of fundamental operation symbols in \(\varphi\) and \(\psi\) are nonempty and identical. In connection with these identities we define in section 1 a construction called an n-clone extension of an algebra for \(0\le n \le |F|\) where n is an integer and we study its properties. For a variety V we denote by V c the variety defined by all clone compatible identities from Id (V). We also define a variety V c,n called the n-clone extension of V. These two varieties are strictly connected. In section 2 under some assumptions we give representations of algebras from V c,n and V c using n-clone extensions of algebras from V. We also find equational bases of these varieties. In section 3 we apply these results to some important varieties. In section 4 we find minimal generics of V c when V is the variety of distributive lattices or the variety of Boolean algebras.

Varieties of distributive lattices with unary operations I

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 eachfμis 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 operationsfμ. 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.

On the Algebraization of Some Gentzen Systems1

In this paper we study the algebraization of two Gentzen systems, both of them generating the implication-less fragment of the intuitionistic propositional calculus. We prove that they are algebraizable, the variety of pseudocomplemented distributive lattices being an equivalent algebraic semantics for them, in the sense that their Gentzen deduction and the equational deduction over this variety are interpretable in one another, these interpretations being essentially inverse to one another. As a consequence, the consistent deductive systems that satisfy the properties of Conjunction, Disjunction and Pseudo-Reductio ad Absurdum are described by giving apropriate Gentzen systems for them. All these Gentzen systems are algebraizable, the subvarieties of the variety of pseudocomplemented distributive lattices being their equivalent algebraic semantics respectively. Finally we give a Gentzen system for the conjunction and disjunction fragment of the classical propositional calculus, prove that the variety of distributive lattices is an equivalent algebraic semantics for it and give a Gentzen system, weaker than the latter, the variety of lattices being an equivalent algebraic semantics for it.

Endomorphism monoids of bands

Endomorphisms of any algebra A form a monoid End(A) under their composition. In this paper we investigate the endomorphism monoids of bands, and the question of when End(A) determines A6V within a variety V of bands. An early result on determination by endomorphisms not of algebras but partially ordered sets is due to L.M. Gluskin [7]. It says that two partially ordered sets S and T whose endomorphism semigroups are isomorphic must themselves be either isomorphic or antiisomorphic. Analogous results hold for some small varieties of algebras, for example, for a variety of distributive lattices, see B.M. Schein []9], R. McKenzie and C. Tsinakis [14], and by P. Ribenboim [18]; for distributive p-algebras see [i], for Boolean rings K.D. Magill [13], for semilattices [19], and for Brouwerian semilattices [i0] and [21]. In all of these varieties there exist at most two non-isomorphic algebras with isomorphic endomorphism monoids. B.M. Schein [20] proved that there exist at most four non-isomorphic normal bands with ismorphic endomorphism monoids. Our aim is to generalize the results of B.M. Schein [19,20] to larger