Starting from Boolean algebras of languages closed under quotients and using duality theoretic insights, we derive the notion of Boolean spaces with internal monoids as recognisers for arbitrary formal languages of finite words over finite alphabets. This leads to recognisers and syntactic spaces equivalent to those proposed in [8], albeit in a setting that is well-suited for applying existing ...

The daseinisation is a mapping from an orthomodular lattice in ordinary quantum theory into Heyting algebra topos theory. While distributivity does not always hold lattices, it algebras. We investigate the conditions under which negations and meets are preserved by daseinisation, condition that any element transformed through corresponds to original lattice. show these equivalent, that, only ca...

The problem of imbedding an abstract distributive lattice in a Boolean algebra by an algebraic extension was suggested to the writer by M. H. Stone in 1933. Hausdorfff had already given a solution of this problem for the case where the given distributive lattice was a ring of point sets. A solution for the abstract case was presented by the writer to the Harvard Mathematical Colloquium (1934), ...

All Boolean variables here range over the two element set {−1, 1}. Given n Boolean variables x1, . . . , xn, a non-monotone MAJORITY gate (in the variables xi) is a Boolean function whose value is the sign of ∑n i=1 ixi, where each i is either 1 or −1. The COMPARISON function is the Boolean function of two n-bits integers X and Y whose value is −1 iff X ≥ Y . We construct an explicit sparse pol...

Boolean problems are conventionally formulated and solved using the algebraic framework originally proposed by George Boole for application to mathematical logic problems. Boole’s algebra was used by Claude Shannon for modeling networks of switching relays and later more general manipulation of information in the form of binary digits, or bits. An alternative approach is to model data as an ele...

We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin...

The finite model property of the variety of S-algebras was proved by X. Caicedo using Kripke model techniques of the associated calculus. A more algebraic proof, but still strongly based on Kripke model ideas, was given by Muravitskii. In this article we give a purely algebraic proof for the finite model property which is strongly based on the fact that for every element x in a S-algebra the in...