Boolean Algebra In Terms Research Articles

Semisimple varieties of implication zroupoids

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein (Trans Am Math Soc 36:876---884, 1934) gave a system of axioms for Boolean algebras in terms of implication only. Though his original axioms were not equational, a quick look at his axioms would reveal that if one adds a constant, then it is not hard to translate his system of axioms into an equational one. Recently, in 2012, the second author of this paper extended this modified Bernstein's theorem to De Morgan algebras (see Sankappanavar, Sci Math Jpn 75(1):21---50, 2012). Indeed, it is shown in Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties, $$\mathbf {DM}$$DM, $$\mathbf {KL}$$KL, and $$\mathbf {BA}$$BA whose defining axioms use only the implication $$\rightarrow $$ź and the constant 0. The fact that the identity, herein called (I), occurs as one of the two axioms in the definition of each of the varieties $$\mathbf {DM}$$DM, $$\mathbf {KL}$$KL and $$\mathbf {BA}$$BA motivated the second author of this paper to introduce, and investigate, the variety $$\mathbf {I}$$I of implication zroupoids, generalizing De Morgan algebras. These investigations are continued by the authors of the present paper in Cornejo and Sankappanavar (Implication zroupoids I, 2015), wherein several new subvarieties of $$\mathbf {I}$$I are introduced and their relationships with each other and with the varieties studied in Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) are explored. The present paper is a continuation of Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) and Cornejo and Sankappanavar (Implication zroupoids I, 2015). The main purpose of this paper is to determine the simple algebras in $$\mathbf {I}$$I. It is shown that there are exactly five (nontrivial) simple algebras in $$\mathbf {I}$$I. From this description we deduce that the semisimple subvarieties of $$\mathbf {I}$$I are precisely the subvarieties of the variety generated by these simple I-zroupoids and that they are locally finite. It also follows that the lattice of semisimple subvarieties of $$\mathbf {I}$$I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.

Boolean sets, skew Boolean algebras and a non-commutative Stone duality

We describe right-hand skew Boolean algebras in terms of a class of presheaves of sets over Boolean algebras called Boolean sets, and prove a duality theorem between Boolean sets and étalé spaces over Boolean spaces.

Sufficient Conditions for Decidability of Boolean Algebras

We study the decidability of Boolean algebras in terms of computability of canonical ideals and prove the sufficiency of conditions in the case of Boolean algebras of all elementary characteristics. Bibliography: 11 titles.

The minimality of certain decidability conditions for Boolean algebras

We complete our study of the decidability of Boolean algebras in terms of computability of a certain sequence of canonical ideals. We present a proof of the minimality of the conditions obtained for the decidability of Boolean algebras of all elementary characteristics.

$\delta$-ideals in pseudo-complemented distributive lattices

The concept of $\delta$-ideals is introduced in a pseudo-complemented distributive lattice and some properties of these ideals are studied. Stone lattices are characterized in terms of $\delta$-ideals. A set of equivalent conditions is obtained to characterize a Boolean algebra in terms of $\delta$-ideals. Finally, some properties of $\delta$-ideals are studied with respect to homomorphisms and filter congruences.

Boolean Algebra of C-Algebras

A C- algebra is the algebraic form of the 3-valued conditional logic, which was introduced by F. Guzman and C.C. Squier in 1990. In this paper, some equivalent conditions for a C- algebra to become a boolean algebra in terms of congruences are given. It is proved that the set of all central elements B(A) is isomorphic to the Boolean algebra

Lattice-valued convergence spaces and regularity

We define a regularity axiom for lattice-valued convergence spaces where the lattice is a complete Heyting algebra. To this end, we generalize the characterization of regularity by a “dual form” of a diagonal condition. We show that our axiom ensures that a regular T1-space is separated and that regularity is preserved under initial constructions. Further we present an extension theorem for a continuous mapping from a subspace to a regular space. A characterization in the restricted case that the lattice is a complete Boolean algebra in terms of the closure of an L-filter is given.

κ-stationary subsets of , infinitary games, and distributive laws in Boolean algebras

AbstractWe characterize the (κ, λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games , when μ ≤ κ ≤ λ and κ<κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, λ, μ with μ ≤ λ, under GCH. In the case when μ ≤ κ ≤ λ and κ<κ = κ, we show that it is necessary to consider whether the κ-stationarity of in the ground model is preserved by . In this vein, we develop the theory of κ-club and κ-stationary subsets of . We also construct Boolean algebras in which Player I wins but the (κ, ∞, κ)-d.1. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.

A Shortest 2-Basis for Boolean Algebra in Terms of the Sheffer Stroke

In this article, we present a short 2-basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2-basis can be shorter. We also prove that the new 2-basis is unique (for its length) up to applications of commutativity. Our proof of the 2-basis was found by using the method of proof sketches and relied on the use of an automated reasoning program.

Short Single Axioms for Boolean Algebra

We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.

Thec-2d-Index of Oriented Matroids

We obtain an explicit method to compute thecd-index of the lattice of regions of an oriented matroid from theab-index of the corresponding lattice of flats. Since thecd-index of the lattice of regions is a polynomial in the ring Z(c,2d), we call it thec-2d-index. As an application we obtain a zonotopal analogue of a conjecture of Stanley: among all zonotopes the cubical lattice has the smallestc-2d-index coefficient-wise. We give a new combinatorial description for thec-2d-index of the cubical lattice and thecd-index of the Boolean algebra in terms of all the permutations in the symmetric groupSn. Finally, we show that only two-thirds of theα(S)'sof the lattice of flats are needed for thec-2d-index computation.

Gelfand theorem implies Stone representation theorem of Boolean rings

Stone Theorem about representing a Boolean algebra in terms of open‐closed subsets of a topological space is a consequence of the Gelfand Theorem about representing a B∗‐ algebra as the algebra of continuous functions on a compact Hausdorff space.

Characterising complete boolean algebras in terms of pure essentialness

We discuss purity and pure essentialness of abelian groups in a topos of sheaves on a locale and show that purity is not a local property. We prove that is divisible if and only if it is pure in every extension, and give an example of a category in winch absolutely pure does not imply divisible. We discuss uniform abelian groups and show that each AU uniform in Ab does not imply that A is uniform in Banaschewski showed that the pure subgroups of are exactly of the type for the different . We show that is essential in if and only if U is dense in , Finally, we characterise as complete boolean algebras the locales for which the only pure and essential subgroup of is .

A topological characterization of complete distributive lattices

An ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ϵ X with x ≰ y there is an increasing clopen set A (i.e. A is closed and open and such that a ϵ A, a ⩽ z implies that z ϵ A) which separates x from y, i.e., x ϵ A and y ≱ A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces. In this paper we shall prove that a lattice L ϵ Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ⊆ X, D is increasing and open implies D ∗ is increasing clopen (where A ∗ denotes the least increasing set which includes A). This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance). We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space.

Stone Lattices. I: Construction Theorems

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

The Lattice of Subalgebras of a Boolean Algebra

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Short Formulations of Boolean Algebra, Using Ring Operations

Special interest has recently attached to formulations of Boolean algebra in terms of ring operations [7], [1]. These axiomatizations have not been as brief as those reached through other modes of approach. The present note will show that the number of axioms when ring operations are used may be as small as in any present version that is not metamathematical, that is, the number of axioms finally employed will be two.

Lee Byrne. Boolean algebra in terms of inclusion. American journal of mathematics, vol. 70 (1918), pp. 139–143.

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Albert Whiteman. Postulates for Boolean algebra in terms of ternary rejection. Bulletin of the American Mathematical Society, vol. 43 (1937), pp. 293–298.

Albert Whiteman. Postulates for Boolean algebra in terms of ternary rejection. Bulletin of the American Mathematical Society, vol. 43 (1937), pp. 293–298.

Postulates for Boolean algebra in terms of Ternary rejection

Postulates for Boolean algebra in terms of Ternary rejection