# zbMATH — the first resource for mathematics

Projectable $$\ell$$-groups and algebras of logic: categorical and algebraic connections. (English) Zbl 1345.06013
Generalized MV algebras, or GMV algebras for short, are “simultaneous generalizations of MV algebras to the noncommutative, unbounded and nonintegral case”, while IGMV algebras are integral GMV algebras. In the paper [N. Galatos and C. Tsinakis, J. Algebra 283, No. 1, 254-291 (2005; Zbl 1063.06008)] it was proved in fact that “the categories of IGMV algebras and of negative cones of $$l$$-groups with a dense nucleus are equivalent”.
This paper answers the natural conjecture “that such an equivalence restricts to an equivalence of the subcategories whose objects are the projectable members of these classes of algebras”. The authors prove that, indeed, “the categories of projectable IGMV algebras and of negative cones of projectable $$l$$-groups with a dense nucleus are equivalent”.
Moreover, by adding the Gödel implication to an IGMV algebra, they introduce the notion of Gödel GMV algebra – as an algebra $$(M,\wedge,\vee,\cdot,\backslash,/,\to,1)$$ of type $$(2,2,2,2,2,2,0)$$ such that $$(M,\wedge,\vee,\cdot,\backslash,/,1)$$ is an IGMV algebra and $$(M,\wedge,\vee,\cdot,\to,1)$$ is a Gödel algebra. And they prove that there is an adjunction between the category of Gödel GMV algebras and a certain category.
Reviewer’s remarks: The readability of the paper is restricted by the use of the divisions (the “Chinese sticks”) $$\backslash$$, $$/$$, instead of the implications $$\to$$, $$\rightsquigarrow$$ ($$y\to z=z/y$$ and $$y\rightsquigarrow z=y\backslash z$$) coming from logic.

##### MSC:
 06F15 Ordered groups 06D35 MV-algebras 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 03G10 Logical aspects of lattices and related structures 06B20 Varieties of lattices 06F05 Ordered semigroups and monoids
Full Text:
##### References:
 [1] Anderson, M.; Conrad, P.; Martinez, J., The lattice of convex ℓ-subgroups of a lattice-ordered group, (Glass, A. M.W.; Holland, W. C., Lattice-Ordered Groups, (1989), Reidel Dordrecht), 105-127 [2] Bahls, P.; Cole, J.; Galatos, N.; Jipsen, P.; Tsinakis, C., Cancellative residuated lattices, Algebra Univers., 50, 1, 83-106, (2003) · Zbl 1092.06012 [3] Balbes, R.; Dwinger, P., Distributive lattices, (1974), University of Missouri Press · Zbl 0321.06012 [4] Blount, K.; Tsinakis, C., The structure of residuated lattices, Int. J. Algebra Comput., 13, 4, 437-461, (2002) · Zbl 1048.06010 [5] Bosbach, B., Residuation groupoids, Results Math., 5, 107-122, (1982) · Zbl 0513.06007 [6] Bosbach, B., Concerning cone algebras, Algebra Univers., 15, 58-66, (1982) · Zbl 0507.06013 [7] Botur, M.; Kühr, J.; Liu, L.; Tsinakis, C., Conrad’s program: from ℓ-groups to algebras of logic, J. Algebra, 450, 173-203, (2016) · Zbl 1337.06006 [8] Chambless, D. A., Representation of the projectable and strongly projectable hulls of a lattice-ordered group, Proc. Am. Math. Soc., 34, 2, 346-350, (1972) · Zbl 0278.06015 [9] Cignoli, R.; D’Ottaviano, I. M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, (1999), Kluwer Dordrecht [10] Conrad, P., The structure of a lattice-ordered group with a finite number of disjoint elements, Mich. Math. J., 7, 171-180, (1960) · Zbl 0103.01501 [11] Conrad, P., Some structure theorems for lattice-ordered groups, Trans. Am. Math. Soc., 99, 212-240, (1961) · Zbl 0099.25401 [12] Conrad, P., The lattice of all convex ℓ-subgroups of a lattice-ordered group, Czechoslov. Math. J., 15, 101-123, (1965) · Zbl 0135.06301 [13] Conrad, P., Lex-subgroups of lattice-ordered groups, Czechoslov. Math. J., 18, 86-103, (1968) · Zbl 0155.05902 [14] Diego, A., Sur LES algébres de Hilbert, Collect. Log. Math., 21, 1-52, (1966) · Zbl 0144.00105 [15] Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated lattices: an algebraic glimpse on substructural logics, (2007), Elsevier Amsterdam · Zbl 1171.03001 [16] Galatos, N.; Tsinakis, T., Generalized MV algebras, J. Algebra, 283, 254-291, (2005) · Zbl 1063.06008 [17] J. Gil-Férez, A. Ledda, C. Tsinakis, Hulls of ordered algebras: projectability, strong projectability and lateral completeness, preprint. · Zbl 1423.06052 [18] Gumm, H. P.; Ursini, A., Ideals in universal algebra, Algebra Univers., 19, 45-54, (1984) · Zbl 0547.08001 [19] Jakubík, J., Retract mappings of projectable MV-algebras, Soft Comput., 4, 27-32, (2000) · Zbl 1005.06007 [20] Jipsen, P.; Tsinakis, C., A survey of residuated lattices, (Martinez, J., Ordered Algebraic Structures, (2002), Kluwer Dordrecht), 19-56 · Zbl 1070.06005 [21] Ledda, A.; Paoli, F.; Tsinakis, C., Lattice-theoretic properties of algebras of logic, J. Pure Appl. Algebra, 218, 1932-1952, (2014) · Zbl 1323.03096 [22] Luxemburg, W.; Zaanen, A., Riesz spaces, vol. 1, (1971), North-Holland Amsterdam and London · Zbl 0231.46014 [23] Metcalfe, G.; Paoli, F.; Tsinakis, C., Ordered algebras and logic, (Hosni, H.; Montagna, F., Probability, Uncertainty, Rationality, (2010), Edizioni della Normale Pisa), 1-85 [24] Tsinakis, C., A unified treatment of certain aspects of the theories of lattice-ordered groups and semiprime rings via Brouwerian lattices and multiplicative lattices, (1979), University of California at Berkeley, Ph.D. dissertation [25] Tsinakis, C., Projectable and strongly projectable lattice-ordered groups, Algebra Univers., 20, 57-76, (1985) · Zbl 0588.06009 [26] Tsinakis, C., Groupable lattices, Commun. Algebra, 23, 13, 4737-4748, (1995) · Zbl 0835.06018 [27] Weglorz, B., Equationally compact algebras I, Fundam. Math., 59, 289-298, (1966) · Zbl 0221.02039 [28] Young, W., From interior algebras to unital ℓ-groups: a unifying treatment of modal residuated lattices, Stud. Log., 103, 2, 265-286, (2015) · Zbl 1379.06006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.