Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 101-114
doi: 10.7151/dmgaa.1066

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Milan Jasem

Department of Mathematics, Faculty of Chemical Technology
Slovak Technical University
Radlinského 9, 812 37 Bratislava, Slovak Republic
e-mail: milian.jasem@stuba.sk


In the paper lattice-ordered monoids and specially normal lattice-ordered monoids which are a generalization of dually residuated lattice-ordered semigroups are investigated. Normal lattice-ordered monoids are metricless normal lattice-ordered autometrized algebras. It is proved that in any lattice-ordered monoid A, a ∈ A and na > 0 for some positive integer n imply a > 0. A necessary and sufficient condition is found for a lattice-ordered monoid A, such that the set I of all invertible elements of A is a convex subset of A and A- ⊆ I, to be the direct product of the lattice-ordered group I and a lattice-ordered semigroup P with the least element 0.

Keywords: lattice-ordered monoid, normal lattice-ordered monoid, dually residuated lattice-ordered semigroup, direct decomposition, polar.

2000 Mathematics Subject Classification: 06F05.


[1]G. Birkhoff, Lattice Theory, Third edition, Amer. Math. Soc., Providence, RI, 1967.
[2]A.C. Choudhury, The doubly distributive m-lattice, Bull. Calcutta. Math. Soc. 47 (1957), 71-74.
[3]L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York 1963.
[4]M. Hansen, Minimal prime ideals in autometrized algebras, Czech. Math. J. 44 (119) (1994), 81-90.
[5]M. Jasem, Weak isometries and direct decompositions of dually residuated lattice-ordered semigroups, Math. Slovaca 43 (1993), 119-136.
[6]T. Kovár, Any DRl-semigroup is the direct product of a commutative l-group and a DRl-semigroup with the least element, Discuss. Math.-Algebra & Stochastic Methods 16 (1996), 99-105.
[7]T. Kovár, A general theory of dually residuated lattice-ordered monoids, Ph.D. Thesis, Palacký Univ., Olomouc 1996.
[8]T. Kovár, Two remarks on dually residuated lattice-ordered semigroups, Math. Slovaca 49 (1999), 17-18.
[9]T. Kovár, On (weak) zero-fixing isometries in dually residuated lattice-ordered semigroups, Math. Slovaca 50 (2000), 123-125.
[10]T. Kovár, Normal autometrized lattice-ordered algebras, Math. Slovaca, 50 (2000), 369-376.
[11]J. Rachnek, Prime ideals in autometrized algebras, Czechoslovak Math. J. 37 (112) (1987), 65-69.
[12]J. Rachnek, Polars in autometrized algebras, Czechoslovak Math. J. 39 (114) (1989), 681-685.
[13]J. Rachnek, Regular ideals in autometrized algebras, Math. Slovaca 40 (1990), 117-122.
[14]J. Rachnek, DRl-semigroups and MV-algebras, Czechoslovak Math. J. 48 (123) (1998), 365-372.
[15]J. Rachnek, MV-algebras are categorically equivalent to a class of DRl1(i)-semigroups, Math. Bohemica 123 (1998), 437-441.
[16]K.L.N. Swamy, Dually residuated lattice-ordered semigroups, Math. Ann. 159 (1965), 105-114.
[17]K.L.N. Swamy, Dually residuated lattice-ordered semigroups. II, Math. Ann. 160 (1965), 64-71.
[18]K.L.N. Swamy, Dually residuated lattice-ordered semigroups. III, Math. Ann. 167 (1966), 71-74.
[19]K.L.N. Swamy and N.P. Rao, Ideals in autometrized algebras, J. Austral. Math. Soc. Ser. A 24 (1977), 362-374. 
[20]K.L.N. Swamy and B.V. Subba Rao, Isometries in dually residuated lattice-ordered semigroups, Math. Sem. Notes Kobe Univ. 8 (1980), 369-379.

Received 26 November 2002
Revised 16 May 2003
Revised 20 November 2003