Homomorphisms of Directed Posets

Chajda, Ivan

doi:10.1142/s1793557108000059

Cited by 6 publications

(4 citation statements)

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“…We are motivated by the fact that to certain relational systems, in particular to directed posets, a certain directoid (see e.g. [3]) or semilattice can be assigned in such a way that a b if and only if a ∨ b = b. Moreover, every homomorphism of a semilattice induces a certain homomorphism of the poset (A, ) as was investigated in [3] or, in the case of (A, E) where E is an equivalence relation on A, in [2].…”

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Groupoids assigned to relational systems

Chajda¹,

Länger²

2013

Math. Bohem.

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By a relational system we mean a couple (A, R) where A is a set and R is a binary relation on A, i.e. R ⊆ A × A. To every directed relational system A = (A, R) we assign a groupoid G(A) = (A, •) on the same base set where xy = y if and only if (x, y) ∈ R. We characterize basic properties of R by means of identities satisfied by G(A) and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.

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“…[3]) or semilattice can be assigned in such a way that a b if and only if a ∨ b = b. Moreover, every homomorphism of a semilattice induces a certain homomorphism of the poset (A, ) as was investigated in [3] or, in the case of (A, E) where E is an equivalence relation on A, in [2]. For quasiordered sets a similar question was solved in [4].…”

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Groupoids assigned to relational systems

Chajda¹,

Länger²

2013

Math. Bohem.

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“…a groupoid with one binary operation. Homomorphisms of such directed posets were already investigated by the first author in [2]. For a bit more general relational structures, so-called quasiordered sets, cone preserving mappings were studied in [4].…”

Section: Introductionmentioning

confidence: 99%

Monotone and cone preserving mappings on posets

Chajda¹,

Länger²

2021

Preprint

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We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to characterize posets in which some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again.

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“…Riguet ([6]) and the classical algebraic approach by A. I. Mal'cev ( [5]). Several properties of equivalences and homomorphisms were treated by the first author in [2] for equivalence systems, in [4] for quasiordered 38 Ivan CHAJDA, Helmut LÄNGER sets and in [3] for posets. It turns out that these results can be extended to arbitrary relational systems.…”

Section: Introductionmentioning

confidence: 99%