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Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Gábor Czédli (2024)

Mathematica Bohemica

Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no M 3 as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset P is said to be JConSPS-representable if there is an SPS lattice L such that P is isomorphic to the poset J ( Con L ) of join-irreducible congruences of L . We prove that if 1 < n ∈ ℕ and...

Relations between some dimensions of semimodular lattices

Andrzej Walendziak (2004)

Czechoslovak Mathematical Journal

The aim of this paper is to present relations between Goldie, hollow and Kurosh-Ore dimensions of semimodular lattices. Relations between Goldie and Kurosh-Ore dimensions of modular lattices were studied by Grzeszczuk, Okiński and Puczyłowski.

Relative conditional expectations on a logic

Oľga Nánásiová, Sylvia Pulmannová (1985)

Aplikace matematiky

In this paper, the authors introduce the notion of conditional expectation of an observable x on a logic with respect to a sublogic, in a state m , relative to an element a of the logic. This conditional expectation is an analogue of the expectation of an integrable function on a probability space.

Relatively additive states on quantum logics

Pavel Pták, Hans Weber (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous...

Relatively complemented ordered sets

Ivan Chajda, Zuzana Morávková (2000)

Discussiones Mathematicae - General Algebra and Applications

We investigate conditions for the existence of relative complements in ordered sets. For relatively complemented ordered sets with 0 we show that each element b ≠ 0 is the least one of the set of all upper bounds of all atoms contained in b.

Representation of algebraic distributive lattices with ℵ1 compact elements as ideal lattices of regular rings.

Friedrich Wehrung (2000)

Publicacions Matemàtiques

We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively,...

Ring-like structures derived from λ -lattices with antitone involutions

Ivan Chajda (2007)

Mathematica Bohemica

Using the concept of the λ -lattice introduced recently by V. Snášel we define λ -lattices with antitone involutions. For them we establish a correspondence to ring-like structures similarly as it was done for ortholattices and pseudorings, for Boolean algebras and Boolean rings or for lattices with an antitone involution and the so-called Boolean quasirings.

Ring-like structures with unique symmetric difference related to quantum logic

Dietmar Dorninger, Helmut Länger, Maciej Maczyński (2001)

Discussiones Mathematicae - General Algebra and Applications

Ring-like quantum structures generalizing Boolean rings and having the property that the terms corresponding to the two normal forms of the symmetric difference in Boolean algebras coincide are investigated. Subclasses of these structures are algebraically characterized and related to quantum logic. In particular, a physical interpretation of the proposed model following Mackey's approach to axiomatic quantum mechanics is given.

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