Decidability in Orthomodular Lattices

We discuss the possibility of automatic simplification of formulas in orthomodular lattices. We describe the principles of a program which decides the validity of equalities and inequalities, as well as implications between them and other important relations significant in quantum mechanics. PACS: 02.10.-v, 02.10.Ab, 02.10.De, 03.65.Fd. AMS Subject classification: 06C15, 03G12, 06B10, 06B25, 81P10.     

Equivalencies, Identities, Symmetric Differences, and Congruencies in Orthomodular Lattices

It is shown that operations of equivalence cannot serve for building algebras which would induce orthomodular lattices as the operations of implication can. Several properties of equivalence operations have been investigated. Distributivity of equivalence terms and several other 3-variable expressions involving equivalence terms have been proved to hold in any orthomodular lattice. Symmetric differences have been shown to reduce to complements of equivalence terms. Some congruence relations related to equivalence operations and symmetric differences have been considered.     

Stone''''s representation theorem in fuzzy topology

In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0 ∈ L is a prime or 1 ∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L-locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L-topological spaces.     

Interpolation operators associated with sub-frame sets

Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the -congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same -congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.

     

A Note on Base Change,Identities Involving τ(n), and a Congruence of Ramanujan

We use the theory of quadratic base change to derive some new identities involving the Ramanujan -function, and show how the Ramanujan congruence (n) ₁₁(n) (mod 691) follows.     

The minimal number of pieces realizing affine congruence by dissection of topological discs

Let be a group of affine transformations of the plane that contains a strict contraction and all translations. It is shown that any two topological discs are congruent dissection with respect to such that only three topological discs are used as pieces of dissection. Two pieces of dissection do not suffice in general even if consists of all affine transformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Congruences of Clone Lattices, II

We continue the study of congruences of clone lattices _A , where A is finite, started in an earlier paper by the author and A. P. Semigrodskikh. We prove that each clone that either contains all unary operations or consists of essentially unary operations forms a one-element class of any non-trivial congruence of _A . As a consequence, we get that _A has the greatest non-trivial congruence provided the lattice is not simple, that _A is directly indecomposable, and that it has neither distributive nor dually distributive elements except for the trivial ones.For |A|>2, no example of a non-trivial congruence is known so far. We exhibit some reasons why such congruences are not easy to find.     

Selfdistributive groupoids of small orders

After enumerating isomorphism types of at most five-element left distributive groupoids, we prove that a distributive groupoid with less than 81 elements is necessarily medial.     

Subalgebras and Homomorphic Images of Algebras Having the CEP and the WCIP

In the present paper we consider algebras satisfying both the congruence extension property (briefly the CEP) and the weak congruence intersection property (WCIP for short). We prove that subalgebras of such algebras have these properties. We deduce that a lattice has the CEP and the WCIP if and only if it is a two-element chain. We also show that the class of all congruence modular algebras with the WCIP is closed under the formation of homomorphic images.