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Keywords:
$DR\ell $-monoid; prime ideal; spectrum
Summary:
Dually residuated lattice-ordered monoids ($DR\ell $-monoids for short) generalize lattice-ordered groups and include for instance also $GMV$-algebras (pseudo $MV$-algebras), a non-commutative extension of $MV$-algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.
References:
[1] R. Balbes, P. Dwinger: Distributive Lattices. University of Missouri Press, Columbia, 1974. MR 0373985
[2] R. L. O. Cignoli, I. M. L. D’Ottawiano, D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht, 2000.
[3] A. Di Nola, G. Georgescu, A. Iorgulescu: Pseudo $BL$-algebras: Part I. Mult. Valued Log. 8 (2002), 673–714. MR 1948853
[4] A. Di Nola, G. Georgescu, A. Iorgulescu: Pseudo $BL$-algebras: Part II. Mult. Valued Log. 8 (2002), 717–750. MR 1948854
[5] R. Engelking: General Topology. PWN, Warszawa, 1977. MR 0500780 | Zbl 0373.54002
[6] G. Georgescu, A. Iorgulescu: Pseudo $MV$-algebras. Mult. Valued Log. 6 (2001), 95–135. MR 1817439
[7] A. M. W. Glass: Partially Ordered Groups. World Scientific, Singapore, 1999. MR 1791008 | Zbl 0933.06010
[8] P. Hájek: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998. MR 1900263
[9] P. Hájek: Basic fuzzy logic and $BL$-algebras. Soft Comput. 2 (1998), 124–128. DOI 10.1007/s005000050043
[10] T. Kovář: A General Theory of Dually Residuated Lattice Ordered Monoids. Ph.D. Thesis, Palacký Univ., Olomouc, 1996.
[11] J. Kühr: Ideals of non-commutative $DR\ell $-monoids. (to appear).
[12] J. Kühr: Prime ideals and polars in $DR\ell $-monoids and pseudo $BL$-algebras. Math. Slovaca 53 (2003), 233–246. MR 2025020
[13] J. Rachůnek: Spectra of autometrized lattice algebras. Math. Bohem. 123 (1998), 87–94. MR 1618727
[14] J. Rachůnek: $MV$-algebras are categorically equivalent to a class of $DR\ell _{1(i)}$-semigroups. Math. Bohem. 123 (1998), 437–441. MR 1667115
[15] J. Rachůnek: A duality between algebras of basic logic and bounded representable $DR\ell $-monoids. Math. Bohem. 126 (2001), 561–569. MR 1970259
[16] J. Rachůnek: A non-commutative generalization of $MV$-algebras. Czechoslovak Math. J. 52 (2002), 255–273. DOI 10.1023/A:1021766309509 | Zbl 1012.06012
[17] J. Rachůnek: Prime spectra of non-commutative generalizations of $MV$-algebras. Algebra Univers. 48 (2002), 151–169. DOI 10.1007/PL00012447 | MR 1929902 | Zbl 1058.06015
[18] J. T. Snodgrass, C. Tsinakis: Finite-valued algebraic lattices. Algebra Univers. 30 (1993), 311–318. MR 1225870
[19] K. L. N. Swamy: Dually residuated lattice ordered semigroups I. Math. Ann. 159 (1965), 105–114. DOI 10.1007/BF01360284 | MR 0183797
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