Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0) → ψ(0) and an induction step that, for generic n, the hypothesis φ(n) → ψ(n) allows one to deduce φ(n+ 1) → ψ(n+ 1). Then it is already true in that arithmetic universe that (∀n)(φ(n) → ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form...

We present and compare two descriptions of state spaces of orthomodular structures (orthoalgebras etc.). Both are based on families of Boolean algebras and their corresponding hypergraphs called Greechie diagrams. The rst approach is based on pasted families of Boolean algebras and gives all orthoalgebras as their pastings. It gives a complete description of the orthoalgebra, as well as its sta...

Posets which are retract of products of chains are characterized by means of two properties: the chain-gap property and the selection property (Rival and Wille, 1981 [8]). Examples of posets with the selection property and not the chain-gap property are easy to find. To date, the Boolean lattice P(ω1)/F in was the sole example of lattice without the selection property [8]. We prove that it does...

Bayesian probability theory is an inference calculus, which originates from a generalization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lattice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry—a generalizati...

D. Lau raised the problem of determining the cardinality of the set of all partial clones of Boolean functions whose total part is a given Boolean clone. The key step in the solution of this problem, which was obtained recently by the authors, was to show that the sublattice of strong partial clones on {0, 1} that contain all total functions preserving the relation ρ0,2 = {(0, 0), (0, 1), (1, 0...

A correlation lattice is an algebra (B,∧,∨, σ, 0, 1) where (B,∧,∨, 0, 1) is a bounded lattice and σ is a dual endomorphism on B which has the property σ(x) = x for a fixed number n, n ∈ 2N + 1 ([3], [4]). This notion generalizes orthomodular lattices, and in the distributive case, Boolean algebras, De Morgan algebras and some classes of Ockham algebras. Moreover, in [4], D. Schweigert and M. Sz...

We highlight a question about binary necklaces, i.e., equivalence classes of binary strings under rotation. Is there a way to choose representatives of the n-bit necklaces so that the subposet of the Boolean lattice induced by those representatives has a symmetric chain decomposition? Alternatively, is the quotient of the Boolean lattice Bn, under the action of the cyclic group Zn, a symmetric ...

Path algebras are additively idempotent semirings and generalize Boolean algebras, fuzzy algebras, distributive lattices and inclines. Thus the Boolean matrices, the fuzzy matrices, the lattice matrices and the incline matrices are prototypical examples of matrices over path algebras. In this paper, generalized fuzzy matrices are considered as matrices over path algebras. Compositions of genera...

Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1, . . . , n} ordered by inclusion. Extending our previous work on a question of Füredi, we show that for any c > 1, there exist functions e(n) ∼ √n/2 and f(n) ∼ c √ n log n and an integer N (depending only on c) such that for all n > N , there is a chain decomposition of the Boolean lattice 2[n] into ( n ⌊n/2⌋ ) ...

An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters. In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual...