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Keywords:
$DR\ell $-monoid; ideal; prime ideal
$DR\ell $-monoid; ideal; prime ideal
Summary:
Lattice-ordered groups, as well as $GMV$-algebras (pseudo $MV$-algebras), are both particular cases of dually residuated lattice-ordered monoids ($DR\ell $-monoids for short). In the paper we study ideals of lower-bounded $DR\ell $-monoids including $GMV$-algebras. Especially, we deal with the connections between ideals of a $DR\ell $-monoid $A$ and ideals of the lattice reduct of $A$.
References:
[1] Cignoli R. L. O., Mundici D., D’Ottaviano I. M. L.: Algebraic Foundations of Many-valued Reasoning. : Kluwer Acad. Publ., Dordrecht-Boston-London. 2000. MR 1786097
[2] Georgescu G., Iorgulescu A.: Pseudo $MV$-algebras. Mult. Valued Log. 6 (2001), 95–135. MR 1817439 | Zbl 1014.06008
[3] Kovář T.: A General Theory of Dually Residuated Lattice Ordered Monoids. Ph.D. Thesis, Palacký University, Olomouc, 1996.
[4] Kühr J.: Ideals of noncommutative $DR\ell $-monoids. Czech. Math. J. (to appear). MR 2121658
[5] Kühr J.: Prime ideals and polars in $DR\ell $-monoids and pseudo $BL$-algebras. Math. Slovaca 53 (2003), 233–246. MR 2025020
[6] Kühr J.: A generalization of $GMV$-algebras. (submitted).
[7] Rachůnek J.: $DR\ell $-semigroups and $MV$-algebras. Czech. Math. J. 48 (1998), 365–372. MR 1624268
[8] Rachůnek J.: $MV$-algebras are categorically equivalent to a class of $DR\ell _{1(i)}$-semigroups. Math. Bohem. 123 (1998), 437–441. MR 1667115
[9] Rachůnek J.: Connections between ideals of non-commutative generalizations of $MV$-algebras and ideals of their underlying lattices. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 40 (2001), 195–200. MR 1904695 | Zbl 1040.06005
[10] Rachůnek J.: A non-commutative generalization of $MV$-algebras. Czech. Math. J. 52 (2002), 255–273. Zbl 1012.06012
[11] Swamy K. L. N.: Dually residuated lattice ordered semigroups. Math. Ann. 159 (1965), 105–114. MR 0183797 | Zbl 0138.02104
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