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On distances and metrics in discrete ordered sets

Stephan Foldes, Sándor Radelecki (2021)

Mathematica Bohemica

Discrete partially ordered sets can be turned into distance spaces in several ways. The distance functions may or may not satisfy the triangle inequality and restrictions of the distance to finite chains may or may not coincide with the natural, difference-of-height distance measured in a chain. It is shown that for semilattices the semimodularity ensures the good behaviour of the distances considered. The Jordan-Dedekind chain condition, which is weaker than semimodularity, is equivalent to the...

On Distributive Fixed-Point Expressions

Helmut Seidl, Damian Niwiński (2010)

RAIRO - Theoretical Informatics and Applications

For every fixed-point expression e of alternation-depth r, we construct a new fixed-point expression e' of alternation-depth 2 and size 𝒪 ( r · | e | ) . Expression e' is equivalent to e whenever operators are distributive and the underlying complete lattice has a co-continuous least upper bound. We alternation-depth but also w.r.t. the increase in size of the resulting expression.

On distributive trices

Kiyomitsu Horiuchi, Andreja Tepavčević (2001)

Discussiones Mathematicae - General Algebra and Applications

A triple-semilattice is an algebra with three binary operations, which is a semilattice in respect of each of them. A trice is a triple-semilattice, satisfying so called roundabout absorption laws. In this paper we investigate distributive trices. We prove that the only subdirectly irreducible distributive trices are the trivial one and a two element one. We also discuss finitely generated free distributive trices and prove that a free distributive trice with two generators has 18 elements.

On duality of submodule lattices

Gábor Czédli, Géza Takách (2000)

Discussiones Mathematicae - General Algebra and Applications

An elementary proof is given for Hutchinson's duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

On extended frames

Jorge Picado (1995)

Commentationes Mathematicae Universitatis Carolinae

Some aspects of extended frames are studied, namely, the behaviour of ideals, covers, admissible systems of covers and uniformities.

On extensions of orthosymmetric lattice bimorphisms

Mohamed Ali Toumi (2013)

Mathematica Bohemica

In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if ( A , * ) is a commutative d -algebra and A 𝔡 its Dedekind completion, then, A 𝔡 can be equipped...

On f -rings that are not formally real

James J. Madden (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Henriksen and Isbell showed in 1962 that some commutative rings admit total orderings that violate equational laws (in the language of lattice-ordered rings) that are satisfied by all totally-ordered fields. In this paper, we review the work of Henriksen and Isbell on this topic, construct and classify some examples that illustrate this phenomenon using the valuation theory of Hion (in the process, answering a question posed in [E]) and, finally, prove that a base for the equational theory of totally-ordered...

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