Finite Boolean Algebra Research Articles

A GLOBAL CONVERGENCE THEOREM IN BOOLEAN ALGEBRA

Robert has established a global convergence theorem in $\{0,1\}^{n}$: If a map $\hat{F}$ from $\{0,1\}^{n}$ to itself is contracting relative to the boolean vector distance $d$, then there exists a positive integer $p \leq n$ such that $\hat{F}^{p}$ is constant. In other words, $\hat{F}$ has a unique fixed point $\xi$ such that for any $x$ in $\{0,1\}^{n}$, we have $\hat{F}^{p}(x) = \xi$. The structure ($\{0,1\},+,\cdot,-,0,1$) may be regarded as the two-element boolean algebra. In this paper, this result is extended to any map $F$ from the product $X$ of $n$ finite boolean algebras to itself.

How True It Is = Who Says It’s True

This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, in this context, of natural language statements, particularly those involving negative knowledge and common knowledge. A case study is presented which brings these problems to the fore. None of the basic material presented here is new to this paper—all has appeared in several papers over many years, by the present author and by others. Much of the development in the literature is more general than here—we have confined things to the Boolean case for simplicity and clarity. Most proofs are omitted, but several of the examples are new. The main virtue of the present paper is its coherent presentation of a systematic point of view—identify the truth value of a formula with the set of those who say the formula is true.

On clones of operations over finite boolean algebras

Using the duality between clones of operations over finite boolean algebras and clones of dual operations over finite sets, in this paper we show that the lattice of clones of operations over a finite boolean algebra is complemented, and we characterize the automorphisms and congruences of the iterative algebra of operations over a finite boolean algebra.

On the structure of the Medvedev lattice

AbstractWe investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size . the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size . and in fact that these big chains occur in every infinite interval. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice.

Sugeno integral in a finite Boolean algebra

The aim of this paper is to provide a representation theorem of the Sugeno integral when the evaluation scale is a finite Boolean algebra. This result is a generalisation of a result proved when the evaluation scale is totally ordered. A major difficulty is the renunciation of the comonotonic functions which have to be replaced by the co-included functions. So we need to care about the properties satisfied by the Sugeno integral when the evaluation scale is a finite totally ordered set. To begin we show that, when the scale is totally ordered, the co-included functions are solely needed to characterise the Sugeno integral and that they are less constraining than the comonotonic functions classically used. Next we focus on finite Boolean algebra, and in this new context we define the Sugeno integral and we present the properties still satisfied. To end this article, we present a representation theorem of the Sugeno integral on a finite Boolean algebra.

Decidability of Cancellative Extension of Monoidal T-norm Based Logic

It is known that the monoidal t-norm based logic (MTL) and many of its schematic extensions are decidable. The usual way how to prove decidability of some schematic extension of MTL is to show that the corresponding class of algebras of truth values has the finite model property (FMP) or the finite embeddability property (FEP). However this method does not work for the extensions whose corresponding classes of algebras have only trivial finite members. Typical examples of such extensions are the product logic and the cancellative extension of MTL (ΠMTL) because the only finite algebras belonging to the corresponding varieties are finite Boolean algebras. The product logic is known to be decidable because of its connection with ordered Abelian groups. However the decidability of ΠMTL was not known. This paper solves this problem.

Algebraic lattices and Boolean algebras

In this paper we establish several equivalent conditions for an algebraic lattice to be a finite Boolean algebra.

ON DUALIZING CLONES AS LAWVERE THEORIES

In this paper we treat clones as Lawvere theories and then dualize them as categories, rather than as single objects of a category of algebras. The approach applies only to some primitive-positive clones, but in return, a structure with somewhat surprising properties is obtained. To illustrate the method, we thoroughly investigate the lattice of clones of operations over a finite boolean algebra.

On possibility and probability measures in finite Boolean algebras

In this paper, we study what possibility and necessity measures are like in a finite Boolean algebra, establishing a classification of possibility measures in this type of algebras. The relation between probability and possibility measures is studied. Some conditions for obtaining probabilities that are coherent with a possibility are given. Lastly, Euclidean distance is used for finding probabilities that are close to a given possibility. Also the closest probability is identified and turns out to be coherent.

On clones, transformation monoids, and finite Boolean algebras

We prove that, for a finite Boolean algebra, there exist only finitely many clones which consist of polynomial functions of the algebra and contain the monoid of its unary polynomial functions.

Jacobi–Trudi Identities for Boolean Tableaux and Ideal-tableaux of Zigzag Posets

Aboolean tableauis an arrayT=(Tij) of the elements of a finite boolean algebra with several rows and infinitely many columns, where the entries increase from left to right and downwards. We study the generating functions for various classes of boolean tableaux. Applying the Gessel–Viennot method to certain nonplanar digraphs, we have determinantal formulas for the generating functions, which are regarded as generalized Jacobi–Trudi identities. By this theorem, we can also deal withideal-tableaux of zigzags, and give some new totally positive matrices.

Some results on abstract commutative ideal theory

Conditions are given for a multiplicative lattice to be a finite Boolean algebra. Multiplicative lattices in which semiprimary elements are primary or in which prime elements are weak meet principal are studied. The lattice of filters of a bounded commutative semilattice are investigated. Finally, we study compactly packed lattices.

Complements in lattices of varieties and equational theories

We investigate various weak conditions ensuring that a lattice be complemented. Using these general results in connection with a famous result due to Lampe, we show that the lattice of all equational theories containing a fixed theory must be complemented if it is lower semicomplemented, thereby answering in the affirmative a question raised by Volkov and Vernikov. Moreover, such a lattice must be a finite Boolean algebra if it has one of the following properties: upper or lower sectionally complemented; incomparably complemented; lower semicomplemented and lower semimodular; or atomistic and upper semimodular.

Bell-type inequalities in horizontal sums of Boolean algebras

We give a necessary and sufficient condition for a Bell-type inequality to hold in a horizontal sum of finitely many finite Boolean algebras.

The dimension of orthomodular posets constructed by pasting Boolean algebras

The main aim of this paper is the calculation of the dimension of certain atomic amalgams. These consist of finite Boolean algebras (blocks) pasted together in such a way that a pair of blocks intersects either trivially in the bounds, or the intersection consists of the bounds, an atom, and its complement.

Power convergent boolean matrices

An n × n binary matrix A is limit dominating if its powers converge to a limit L and A ⩾ L. (The inequality holds entrywise, and arithmetic on all binary matrices is assumed to be Boolean: 1 + 1 = 1.) Limit dominating matrices generalize transitive matrices ( A ⩾ A 2). If A is limit dominating, we show that the power limit L can be found immediately from A and that the residual matrix N = A ⧸ L is nilpotent. Moreover, A k = L+N k for all positive integers k. We prove that v⩾ k⩾ n/( n- v+1), where k is the index of convergence of A (the first integer k such that A k = L), and v is the index of convergence of N. We characterize an extreme class of limit dominating matrices, the idempotent matrices ( A 2= A). In particular, we show that a binary matrix A is idempotent if and only if it is limit dominating and the number of nonzero diagonal blocks in its Frobenius normal form equals its column rank (as a Boolean matrix). Finally, we give natural generalizations to matrices with entries from an arbitrary finite Boolean algebra.

Linear operators preserving invariants of nonbinary boolean matrices

There is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary (i.e. zero-one) matrices. This isomorphism allows many problems concerning nonbinary Boolean matrices to the referred to the binary ease. However, there are some features of the general (i.e. nonbinary) case that have not been mentioned, although they differ somewhat from the binary case. We exhibit characterizations of the linear operators that preserve several invariants of matrices over finite Boolean algebras to illustrate the differences and similarities of the general vs. the binary cases. We employ a canonical form that is useful in applying the isomorphism.

Boolean spectral theory

The fact that there is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary [i.e. (0, 1)] matrices appears to have deterred the presentation of solutions, if not the consideration of problems, concerning general (i.e. nonbinary) Boolean matrices. We try to point out that there are interesting features of the general case that have been lost from view. We reopen the investigation of eigenvalues and their corresponding eigenspaces for matrices over an arbitrary finite Boolean algebra B , answering several questions raised by the pioneering work of D.E. Rutherford and providing concise proofs of some of his results. We also extend a result of Wedderburn characterizing the bases of the vector space of n-tuples over B to a characterization of the bases of all vector spaces over B . Using this characterization of bases, we present a straightforward way to find a basis for the eigenspace of each eigenvalue.

Finite pseudo-Boolean and topological Boolean algebras not having an independent basis of quasiidentities

Finite pseudo-Boolean and topological Boolean algebras not having an independent basis of quasiidentities

Canonical partition theorems for parameter sets

A canonical (i.e., unrestricted) version of the partition theorem for k-parameter sets of Graham and Rothschild ( Trans. Amer. Math. Soc. 159 (1971), 257–291) is proven. Some applications, e.g., canonical versions, of the Rado-Folkman-Sanders theorem and of the partition theorem for finite Boolean algebras are given. Also the Erdös-Rado canonization theorem ( J. London Math. Soc. 25 (1950), 249–255) turns out to be an immediate corollary.