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Keywords:
lattice ordered monoid; ideal; normal ideal; congruence relation; dually residuated lattice ordered monoid
Summary:
In the paper the notion of an ideal of a lattice ordered monoid $A$ is introduced and relations between ideals of $A$ and congruence relations on $A$ are investigated. Further, it is shown that the set of all ideals of a soft lattice ordered monoid or a negatively ordered monoid partially ordered by inclusion is an algebraic Brouwerian lattice.
References:
[1] G. Birkhoff: Lattice Theory. Third edition, Amer. Math. Soc., Providence, 1967. MR 0227053 | Zbl 0153.02501
[2] A. Dvurečenskij: Pseudo MV-algebras are intervals in l-groups. J. Austral. Math. Soc. 72 (2002), 427–445. MR 1902211
[3] G. Georgescu, A. Iorgulescu: Pseudo MV-algebras. Mult.-Valued Log. 6 (2001), 95–135. MR 1817439
[4] G. Grätzer: General Lattice Theory. Akademie-Verlag, Berlin, 1978. MR 0504338
[5] M. Hansen: Minimal prime ideals in autometrized algebras. Czech. Math. J. 44 (1994), 81–90. MR 1257938 | Zbl 0814.06011
[6] M. Jasem: On lattice-ordered monoids. Discuss. Math. Gen. Algebra Appl. 23 (2003), 101–114. MR 2070376 | Zbl 1072.06010
[7] M. Jasem: On polars and direct decompositions of lattice ordered monoids. Contributions to General Algebra 16, Verlag Johannes Heyn, Klagenfurt, 2005, pp. 115–131. MR 2166952 | Zbl 1078.06008
[8] T. Kovář: A General Theory of Dually Residuated Lattice Ordered Monoids. Doctoral Thesis, Palacký Univ., Olomouc, 1996.
[9] J. Kühr: Dually Residuated Lattice Ordered Monoids. Doctoral Thesis, Palacký Univ., Olomouc, 2003. Zbl 1066.06008
[10] J. Kühr: Ideals of non-commutative DRl-monoids. Czech. Math. J. 55 (2005), 97–111.
[11] J. Kühr: Prime ideals and polars in DRl-monoids and pseudo BL-algebras. Math. Slovaca 53 (2003), 233–246. MR 2025020
[12] J. Rachůnek: Prime ideals in autometrized algebras. Czech. Math. J. 37 (1987), 65–69.
[13] K. L. M. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. MR 0183797 | Zbl 0138.02104
[14] D. Šalounová: Lex-ideals of DRl-monoids and GMV-algebras. Math. Slovaca 53 (2003), 321–330. MR 2025465
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