Abstract
An involutive residuated lattice (IRL) is a lattice-ordered monoid possessing residual operations and a dualizing element. We show that a large class of self-dual lattices may be endowed with an IRL structure, and give examples of lattices which fail to admit IRLs with natural algebraic conditions. A classification of all IRLs based on the modular lattices M n is provided.
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Blount, K.: On the structure of residuated lattices. Ph.D. Thesis, Vanderbilt University (1999)
Galatos, N.: Varieties of residuated lattices. Ph.D. Thesis, Vanderbilt University (2003)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier Science (2007)
Galatos, N., Olson, J.S., Raftery, J.: Irreducible semilattices. Rep. Math. Logic 43, 85–108 (2008)
Galatos, N., Raftery, J.: A category equivalence for odd Sugihara monoids. J. Pure Appl. Algebra 216, 2177–2192 (2012)
Galatos, N., Raftery, J.: Adding involution to residuated structures. Stud. Logica 77, 181–207 (2004)
Kamara, M., Schweigert, D.: Geometric correlation lattices. Demonstratio Math. 22, 167–177 (1989)
Jipsen, P.: From semirings to residuated Kleene lattices. Stud. Logica 76, 291–303 (2004)
Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)
Olson, J.S.: Fixed elements in involutive residuated lattices. Algebra Univers. 65, 9–19 (2011)
Wild, M.: Three notes on distributive lattices. Adv. Appl. Math. 20, 44–49 (1998)
Wille, A.: Residuated structures with involution. Ph.D. Thesis, Technische Universität Darmstadt, Shaker Verlag (2006)
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Olson, J.S. Involutive Residuated Lattices Based on Modular and Distributive Lattices. Order 31, 373–389 (2014). https://doi.org/10.1007/s11083-013-9307-3
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DOI: https://doi.org/10.1007/s11083-013-9307-3