Pataki, Gergely; Száz, Árpád Characterizations of nonexpansive multipliers on partially ordered sets. (English) Zbl 0991.06001 Math. Slovaca 51, No. 4, 371-382 (2001). 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)]. Reviewer: Jarmila Hedlíková (Bratislava) Cited in 1 Document MSC: 06A06 Partial orders, general 06A12 Semilattices Keywords:partially ordered set; semilattice; nonexpansive multiplier; quasi-interior operator Citations:Zbl 0078.02001; Zbl 0112.01901; Zbl 0078.02002; Zbl 0113.01901; Zbl 0318.06016 PDFBibTeX XMLCite \textit{G. Pataki} and \textit{Á. Száz}, Math. Slovaca 51, No. 4, 371--382 (2001; Zbl 0991.06001) Full Text: EuDML References: [1] BIRKHOFF G.: Lattice Theory. Amer. Math. Soc, Providence, RI, 1973. · Zbl 0063.00402 [2] BRAINERD B.-LAMBEK J.: On the ring of quotients of a Boolean ring. Canad. Math. Bull. 2 (1959), 25-29. · Zbl 0085.26104 · doi:10.4153/CMB-1959-006-x [3] CORNISH W. H.: The multiplier extension of a distributive lattice. J. Algebra 32 (1974), 339-355. · Zbl 0318.06016 · doi:10.1016/0021-8693(74)90143-4 [4] KOLIBIAR M.: Bemerkungen über translationen der Verbände. Acta Fac. Rerum Natur. Univ. Comenian. Math. 5 (1961), 455-458. · Zbl 0113.01901 [5] KOVACS I.-SZAZ A.: Characterizations of effective sets and nonexpansive multipliers in conditionally complete and infinitely distributive partially ordered sets. Acta Math. Acad. Paedagog. Nyhazi. 17 (2001), 61-69 · Zbl 1001.06001 [6] SCHMID J.: Multipliers on distributive lattices and rings of quotients I. Houston J. Math. 6 (1980), 401-425. · Zbl 0501.06008 [7] SZASZ G.: Die Translationen der Halbverbande. Acta Sci. Math. (Szeged) 17 (1956), 165-169. [8] SZASZ G.: Translationen der Verbande. Acta Fac. Rerum Natur. Univ. Comenian. Math. 5 (1961), 449-453. · Zbl 0112.01901 [9] SZASZ G.-SZENDREI J.: Über die Translation der Halbverbande. Acta Sci. Math. (Szeged) 18 (1957), 44-47. · Zbl 0078.02002 [10] SZAZ A.: Partial multipliers on partially ordered sets. Technical Report (Inst. Math. Inf., Univ. Debrecen) 98/8 (1998), 1-28. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.