Variety Of Boolean Algebras Research Articles

New perspectives on semi-primal varieties

We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.

A Note on 3×3-valued Łukasiewicz Algebras with Negation

In 2004, C. Sanza, with the purpose of legitimizing the study of \(n\times m\)-valued Łukasiewicz algebras with negation (or \(\mathbf{NS}_{n\times m}\)-algebras) introduced \(3 \times 3\)-valued Łukasiewicz algebras with negation. Despite the various results obtained about \(\mathbf{NS}_{n\times m}\)-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \(NS_{3 \times 3}\)-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \(\Lambda(\mathbf{NS}_{3\times 3})\) of all subvarieties of \(\mathbf{NS}_{3\times 3}\) and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \(\mathbf{NS}_{3\times 3}\).

Classical Logic with n Truth Values as a Symmetric Many-Valued Logic

We introduce Boolean-like algebras of dimension n (n{mathrm {BA}}s) having n constants {{{mathsf {e}}}}_1,ldots ,{{{mathsf {e}}}}_n, and an (n+1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n{mathrm {BA}}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n{mathrm {BA}}s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n{mathrm {CL}}, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n{mathrm {CL}}, and, via the embeddings, in all the finite tabular logics.

Atom-canonicity in varieties of cylindric algebras with applications to omitting types in multi-modal logic

Fix and let denote the class of cylindric algebras of dimension n. Roughly, is the algebraic counterpart of the proof theory of first-order logic restricted to the first n variables which we denote by . The variety of representable s reflects algebraically the semantics of . A variety of Boolean algebras with operators is atom-canonical, if whenever is atomic, then its Dedekind–MacNeille completion is also in . We show using a so-called blow up and blur construction that for any , any variety of the form containing (and including) (when ) is not atom-canonical. We show that a restricted form of that the celebrated Henkin–Orey omitting types theorem, which we refer to below as Vaught's Theorem (), fails dramatically for even if we allow only so-called m-square models for . We deduce that many multi-modal logics, like , the m clique guarded fragments and packed fragments of are not Sahlqvist. In contrast, we prove a positive for theories with respect to usual semantics, by imposing extra conditions (such as quantifier elimination) on possibly uncountable theories considered and/or the non-principal types omitted (that are also allowed to be possibly uncountable) such as completeness.ast is a maximality condition delineating the edge of an independent statement to a provable one. is shown that the maximality condition cannot be removed even to prove the weaker for uncountable theories; and this construction is used to reprove a celebrated result of Hirsch and Hodkinson, namely, that the class of completely representable s is not el.

Uniform interpolation and coherence

A variety V is said to be coherent if every finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that coherence corresponds to a key ingredient of uniform deductive interpolation for equational consequence in V: the property that any compact congruence on a finitely generated free algebra of V restricted to a free algebra over fewer generators is compact. A general criterion is derived for establishing failures of coherence, and hence also of uniform deductive interpolation. The criterion is then applied in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (including varieties for the modal logic K and KT), double-Heyting algebras, residuated lattices, and lattices.

Categorical equivalence and the Ramsey property for finite powers of a primal algebra

In this paper, we investigate the best known and most important example of a categorical equivalence in algebra, that between the variety of boolean algebras and any variety generated by a single primal algebra. We consider this equivalence in the context of Kechris-Pestov-Todorcevic correspondence, a surprising correspondence between model theory, combinatorics and topological dynamics. We show that relevant combinatorial properties (such as the amalgamation property, Ramsey property and ordering property) carry over from a category to an equivalent category. We then use these results to show that the category whose objects are isomorphic copies of finite powers of a primal algebra $${\mathcal{A}}$$ together with a particular linear ordering <, and whose morphisms are embeddings, is a Ramsey age (and hence a Fraisse age). By the Kechris-Pestov-Todorcevic correspondence, we then infer that the automorphism group of its Fraisse limit is extremely amenable. This correspondence also enables us to compute the universal minimal flow of the Fraisse limit of the class $${{\bf V}_{fin} \mathcal{(A)}}$$ whose objects are isomorphic copies of finite powers of a primal algebra $${\mathcal{A}}$$ and whose morphisms are embeddings.

Symmetric operators on modal pseudocomplemented De Morgan algebras

Modal pseudocomplemented De Morgan algebras are pseudocomplemented De Morgan algebras which satisfy the additional equation |$\mbox{(tm) }\, x\vee \sim x \leq x\vee x^\ast.$| There is a connection between these algebras and A. Monteiro’s tetravalent modal algebras. It is worth mentioning that these algebras have been studied previously using topological tools. In the present work, we enrich the language of modal pseudocomplemented De Morgan algebras by adding an automorphism of period |$2$|⁠, giving rise to a new algebraic structure which we call |$\cal S$|-algebras. From the logical point of view, we are enriching the original language with a new involutive unary connective. Among other things, it is proved that |$\cal S$| is a semisimple, locally finite variety. Besides, the generating algebras of this variety are determined. The structure of the free |$\cal S$|-algebra with a finite number of generators is studied, and a formula for computing its size is given. Finally, the lattice |$\Lambda({\cal S})$| is exhibited showing that the varieties of Boolean algebras, involutive Boolean algebras and three-valued Łukasiewicz algebras are proper subvarieties of |${\cal S}$|⁠.

On varieties of basic algebras

It is known that the variety of commutative basic algebras, the variety of MV-algebras, the variety of orthomodular lattices and hence also the variety of Boolean algebras are proper subvarieties of the variety of basic algebras. In the paper several possible axiomatizations of the variety of basic algebras are provided. The variety of symmetric basic algebras is introduced and shown to be a proper subvariety of the variety of commutative basic algebras, and whose members have the underlying lattices distributive. It is proved that commutative basic algebras can be described as residuated $$l$$ l -groupoids.

SUPER-BOOLEAN FUNCTIONS AND FREE BOOLEAN QUASILATTICES

A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables.

Atomless varieties

AbstractWe define a nontrivial variety of boolean algebras with operators such that every member of the variety is atomless. This shows that not every variety of boolean algebras with operators is generated by its atomic members, and thus establishes a strong incompleteness result in (multi-)modal logic.

Persistence and Atomic Generation for Varieties of Boolean Algebras with Operators

A variety V of Boolean algebras with operators is singleton-persistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atom-canonical if it contains the complex algebra of the atom structure of any of the atomic members of V. This paper explores relationships between these "persistence" properties and questions of whether V is generated by its complex algebras or its atomic members, or is closed under canonical embedding algebras or completions. It also develops a general theory of when operations involving complex algebras lead to the construction of elementary classes of relational structures.

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.

Atom structures and Sahlqvist equations

This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra $ \frak U $ is determined by its atom structure $ \frak {Ut \,\, U} $ . We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary to the positive result and a recent result by I. Hodkinson, we prove that the variety RRA of representable relation algebras, although canonical, cannot be axiomatized by Sahlqvist equations.

On Gentzen Systems Associated with the Finite Linear MV-algebras

In this paper we obtain a characterization of the algebraizability of an m-dimensional Gentzen system in line with the characterization obtained for m-dimensional deductive systems and the characterization of 2-dimensional Gentzen systems. We also prove that if S(m) is the finite linear MV-algebraof m elements, then the m-dimensional Gentzen system obtained by using the sequent calculi associated with S(m) is equivalent to the m-valued Łukasiewicz logic Ł m and to the equational consequence relation associated with S(m). Taking the two-element Boolean algebra we obtain the expected result concerning the relationship between the sequent calculus LK, the Classical Prepositional Calculus and the variety of Boolean algebras.

Dimensions of Cantor and post varieties

A dimension of a finitely based variety V of algebras is the greatest length of a basis (that is, an independent generating set) for the SC-theory SC(V) with the strong Mal'tsev conditions satisfied in V. A dimension is said to be infinite if the lengths of bases in SC(V) are unbounded. We prove that the dimension of a Cantor variety Cm,n in the general form, i.e., with n>m≥1, is infinite. The algorithm of constructing a basis of any given length in SC(Cm,n) is presented. By contrast, any Post variety Pn generated by a primal algebra of order n>1 is shown to have a finite dimension not exceeding the number of maximal subalgebras in the iterative Post algebra over the set {0,1,…,n−1}. Specifically, the dimension of the variety of Boolean algebras is at most four.

On Finite-Valued Propositional Logical Calculi

In this paper we describe, in a purely algebraic language, truth-complete finite-valued propositional logical calculi extending the classical Boolean calculus. We also give a new proof of the Completeness Theorem for such calculi. We investigate the quasi-varieties of algebras playing an analogous role in the theory of these finite-valued logics to the role played by the variety of Boolean algebras in classical logic.

On Ockham algebras: Congruence lattices and subdirectly irreducible algebras

Distributive bounded lattices with a dual homomorphism as unary operation, called Ockham algebras, were firstly studied by Berman (1977). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras are some of the well known subvarieties of Ockham algebra. In this paper, new results about the congruence lattice of Ockham algebras are given. From these results and Urquhart's representation theorem for Ockham algebras a complete characterization of the subdirectly irreducible Ockham algebras is obtained. These results are particularized for a large number of subvarieties of Ockham algebras. For these subvarieties a full description of their subdirectly irreducible algebras is given as well.

What the finitization problem is not

0. Introduction. The finitization problem is one of the problems considered important in algebraic logic. Since it is often misunderstood, it seems desirable to try to develop a better understanding of what it is and what it is not about. The finitization problem is (at least partly) motivated by the following observations (1)–(3): (1) The natural algebraic counterpart of propositional logic is the variety of Boolean algebras which is axiomatizable by finitely many equations. (2) The natural algebraic counterpart of first-order logic Lωω is the variety RCAω of representable cylindric algebras which is very far from being axiomatizable by a finite schema of equations, cf. [HMT], Theorem 4.1.3, and Andreka [A91]. (3) The algebraic counterpart of Lωω without equality is the variety RQPAω of representable quasi-polyadic algebras which is also not axiomatizable by a finite schema, see Sain–Thompson [ST]. (Similar negative results apply to the finitevariable fragments of Lωω. The algebraic counterparts here are finite-dimensional representable cylindric or polyadic algebras and relation algebras.) The negative results (2) and (3) do have purely logical consequences motivating the (nonalgebraic) logician to look into the question, cf. e.g. §4 of Sain [S87], Simon [S90], [S91], Venema [V90], [V92], Nemeti [N91], but see also the present Appendix A. Observations (2) and (3) above are in contrast with (1). So there seems to be a sharp contrast between the behaviour of propositional logic and quantifier logics. It is natural to ask whether the nonfinitizability in (2) and (3) is an unavoidable

On L∞κ-free Boolean algebras

We study L ∞κ-freeness in the variety of Boolean algebras. It is shown that some of the theorems on L ∞κ-free algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean algebras. But we also investigate properties such as the ccc of L ∞κ-free Boolean algebras which have no counterpart in the varieties above.

On the löwenheim numbers for the skeletons of varieties of boolean algebras

On the löwenheim numbers for the skeletons of varieties of boolean algebras