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A binary operation-based representation of a lattice

Mourad Yettou, Abdelaziz Amroune, Lemnaouar Zedam (2019)


In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new...

A note on uniserial loops

Jaroslav Ježek, Tomáš Kepka (2010)

Commentationes Mathematicae Universitatis Carolinae

All ordinal numbers α with the following property are found: there exists a loop such that its subloops form a chain of ordinal type α .

A relational semantics for the logic of bounded lattices

Luciano J. González (2019)

Mathematica Bohemica

This paper aims to propose a complete relational semantics for the so-called logic of bounded lattices, and prove a completeness theorem with regard to a class of two-sorted frames that is dually equivalent (categorically) to the variety of bounded lattices.

A representation theorem for certain Boolean lattices.

José Ríos Montes (1988)

Publicacions Matemàtiques

Let R be an associative ring with 1 and R-tors the somplete Brouwerian lattice of all hereditary torsion theories on the category of left R-modules. A well known result asserts that R is a left semiartinian ring iff R-tors is a complete atomic Boolean lattice. In this note we prove that if L is a complete atomic Boolean lattice then there exists a left semiartinian ring R such that L is lattice-isomorphic to R-tors.

Classification systems and their lattice

Sándor Radeleczki (2002)

Discussiones Mathematicae - General Algebra and Applications

We define and study classification systems in an arbitrary CJ-generated complete lattice L. Introducing a partial order among the classification systems of L, we obtain a complete lattice denoted by Cls(L). By using the elements of the classification systems, another lattice is also constructed: the box lattice B(L) of L. We show that B(L) is an atomistic complete lattice, moreover Cls(L)=Cls(B(L)). If B(L) is a pseudocomplemented lattice, then every classification system of L is independent and...

Congruence lattices of intransitive G-Sets and flat M-Sets

Steve Seif (2013)

Commentationes Mathematicae Universitatis Carolinae

An M-Set is a unary algebra X , M whose set M of operations is a monoid of transformations of X ; X , M is a G-Set if M is a group. A lattice L is said to be represented by an M-Set X , M if the congruence lattice of X , M is isomorphic to L . Given an algebraic lattice L , an invariant Π ( L ) is introduced here. Π ( L ) provides substantial information about properties common to all representations of L by intransitive G-Sets. Π ( L ) is a sublattice of L (possibly isomorphic to the trivial lattice), a Π -product lattice. A Π -product...

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