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Characterizations of nonexpansive multipliers on partially ordered sets. (English) Zbl 0991.06001

A function \(f\) from a subset \(\mathcal D\) of a poset (partially ordered set) \(\mathcal A\) into \(\mathcal A\) is nonexpansive if \(f(d)\leq d\) for all \(d \in \mathcal D\). A function \(f\) from a subset \(\mathcal D\) of a poset \(\mathcal A\) into \(\mathcal A\) is a multiplier if \(f(d)\wedge e=f(e)\wedge d\) for all \(d,e\in \mathcal D\). A function \(f\) from a subset \(\mathcal D\) of a poset \(\mathcal A\) into a poset \(\mathcal B\) is nondecreasing if \(f(d)\leq f(e)\) for all \(d,e\in \mathcal D\) with \(d\leq e\). A function \(f\) from a subset \(\mathcal D\) of a set \(\mathcal A\) into \(\mathcal A\) is quasi-idempotent if \(f\bigl (f(d)\bigr)=f(d)\) for all \(d\in \mathcal D\) with \(f(d)\in \mathcal D\). A quasi-idempotent function \(f\) from a subset \(\mathcal D\) of a set \(\mathcal A\) into \(\mathcal A\) is idempotent if \(f[D]\subseteq D\). A function \(f\) from a subset \(\mathcal D\) of a poset \(\mathcal A\) into \(\mathcal A\) is a quasi-interior operator if it is nonexpansive, nondecreasing and quasi-idempotent. A quasi-interior operator \(f\) from a subset \(\mathcal D\) of a poset \(\mathcal A\) into \(\mathcal A\) is an interior operator if \(f[\mathcal D]\subseteq D\). A nonempty subset \(\mathcal B\) of a poset \(\mathcal A\) is a semilattice in \(\mathcal A\) if \(d \wedge e\) exists in \(\mathcal A\) and belongs to \(\mathcal B\) for all \(d,e\in \mathcal B\). A function \(f\) from a subset \(\mathcal D\) of a poset \(\mathcal A\) into a poset \(\mathcal B\) is quasi-multiplicative if \(f(d\wedge e)=f(d)\wedge f(e)\) for all \(d,e\in \mathcal D\) such that \(d\wedge e\) exists in \(\mathcal A\) and belongs to \(\mathcal D\). A quasi-multiplicative function \(f\) from a subset \(\mathcal D\) of a poset \(\mathcal A\) into a poset \(\mathcal B\) is multiplicative if \(\mathcal D\) is a semilattice in \(\mathcal A\).
The authors present basic characterizations of the defined sorts of functions and operators and establish several relationships between them, thus obtaining an extension and a supplementation of some previous results of G. Szász [“Die Translationen der Halbverbände”, Acta Sci. Math. 17, 165-169 (1956; Zbl 0078.02001); “Translationen der Verbände”, Acta Fac. Rer. Nat. Univ. Comenianae, Math. 5, 449-453 (1961; Zbl 0112.01901)], G. Szász and J. Szendrei [“Über die Translation der Halbverbände”, Acta Sci. Math. 18, 44-47 (1957; Zbl 0078.02002)], M. Kolibiar [“Bemerkungen über Translationen der Verbände”, Acta Fac. Rer. Nat. Univ. Comenianae, Math. 5, 455-458 (1961; Zbl 0113.01901)], W. H. Cornish [“The multiplier extension of a distributive lattice”, J. Algebra 32, 339-355 (1974; Zbl 0318.06016)] and Á. Szász [“Partial multipliers on partially ordered sets”, Technical Report, Inst. Math. Inf., Univ. Debrecen 98/8, 1-28 (1998)].

MSC:

06A06 Partial orders, general
06A12 Semilattices
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References:

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