Geodesic
Interior and closure operators on bounded residuated lattice ordered monoids
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 345-357

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$GMV$-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior $GMV$-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on $DRl$-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on $GMV$-algebras.
$GMV$-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior $GMV$-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on $DRl$-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on $GMV$-algebras.
Classification : 03G25, 06D35, 06F05
Keywords: $GMV$-algebra; $DRl$-monoid; filter 
Švrček, Filip. Interior and closure operators on bounded residuated lattice ordered monoids. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 345-357. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a3/
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[1] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis: Cancellative residuated lattices. Alg. Univ. 50 (2003), 83–106. | DOI | MR

[2] K. Blount and C. Tsinakis: The structure of residuated lattices. Intern. J. Alg. Comp. 13 (2003), 437–461. | DOI | MR

[3] R. O. L. Cignoli, I. M. L. D’Ottaviano and D. Mundici: Algebraic Foundations of Many-Valued Reasoning. Kluwer Acad. Publ., Dordrecht-Boston-London, 2000. | MR

[4] A. Dvurečenskij: States on pseudo $MV$-algebras. Studia Logica 68 (2001), 301–327. | DOI | MR

[5] A. Dvurečenskij: Every linear pseudo $BL$-algebra admits a state. Soft Computing (2006).

[6] A. Dvurečenskij and M. Hyčko: On the existence of states for linear pseudo $BL$-algebras. Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia 53 (2005), 93–110. | MR

[7] A. Dvurečenskij and S. Pulmannová: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht-Boston-London, 2000. | MR

[8] A. Dvurečenskij and J. Rachůnek: On Riečan and Bosbach states for bounded $Rl$-monoids. (to appear).

[9] A. Dvurečenskij and J. Rachůnek: Probabilistic averaging in bounded $Rl$-monoids. Semigroup Forum 72 (2006), 190–206. | MR

[10] G. Georgescu and A. Iorgulescu: Pseudo-$MV$-algebras. Multiple Valued Logic 6 (2001), 95–135. | MR

[11] A. di Nola, G. Georgescu and A. Iorgulescu: Pseudo-$BL$-algebras I. Multiple Valued Logic 8 (2002), 673–714. | MR

[12] A. di Nola, G. Georgescu and A. Iorgulescu: Pseudo-$BL$-algebras II. Multiple Valued Logic 8 (2002), 715–750. | MR

[13] P. Hájek: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. | MR

[14] P. Jipsen and C. Tsinakis: A survey of residuated lattices. Ordered algebraic structures (ed. J. Martinez), Kluwer Acad. Publ. Dordrecht, 2002, pp. 19–56. | MR

[15] T. Kovář: A General Theory of Dually Residuated Lattice Ordered Monoids. Ph.D. Thesis Palacký University, Olomouc, 1996.

[16] J. Kühr: Dually Residuated Lattice Ordered Monoids. Ph.D. Thesis, Palacký Univ., Olomouc, 2003. | MR

[17] J. Kühr: Remarks on ideals in lower-bounded dually residuated lattice-ordered monoids. Acta Univ. Palacki. Olomouc, Mathematica 43 (2004), 105–112. | MR

[18] J. Kühr: Ideals of noncommutative ${\mathcal{D}}Rl$-monoids. Czech. Math. J. 55 (2002), 97–111. | MR

[19] J. Rachůnek: $DRl$-semigroups and $MV$-algebras. Czech. Math. J. 48 (1998), 365–372. | DOI

[20] J. Rachůnek: $MV$-algebras are categorically equivalent to a class of $DRl_{1(i)}$-semigroups. Math. Bohem. 123 (1998), 437–441. | MR

[21] J. Rachůnek: A duality between algebras of basic logic and bounded representable $DRl$-monoids. Math. Bohem. 126 (2001), 561–569. | MR

[22] J. Rachůnek: A non-commutative generalization of $MV$-algebras. Czech. Math. J. 52 (2002), 255–273. | DOI | MR

[23] J. Rachůnek and V. Slezák: Bounded dually residuated lattice ordered monoids as a generalization of fuzzy structures. Math. Slovaca 56 (2006), 223–233. | MR

[24] J. Rachůnek and D. Šalounová: Local bounded commutative residuated $l$-monoids. (to appear). | MR

[25] J. Rachůnek and F. Švrček: $MV$-algebras with additive closure operators. Acta Univ. Palacki., Mathematica 39 (2000), 183–189.

[26] F. Švrček: Operators on $GMV$-algebras. Math. Bohem. 129 (2004), 337–347. | MR

[27] H. Rasiowa and R. Sikorski: The Mathematics of Metamathematics. Panstw. Wyd. Nauk., Warszawa, 1963. | MR

[28] K. L. N. Swamy: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. | DOI | MR | Zbl

[29] E. Turunen: Mathematics Behind Fuzzy Logic. Physica-Verlag, Heidelberg-New York, 1999. | MR | Zbl