4 research outputs found

    On a Definition of a Variety of Monadic ℓ-Groups

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    In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL 0 of integral residuated lattices with bottom, which generalize MV-algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor K∙, motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV-algebras and the corresponding category MV∙ of monadic MV-algebras induced by “Kalman’s functor” K∙. Moreover, we extend the construction to ℓ-groups introducing the new category of monadic ℓ-groups together with a functor Γ♯, that is “parallel” to the well known functor Γ between ℓ and MV-algebras.Facultad de Ciencias Exacta

    Weakening Relation Algebras and FL\u3csup\u3e2\u3c/sup\u3e-algebras

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    FL2-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL2-algebras. We prove that the congruences of FL2-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras. For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class S(dRA) of subalgebras of {p/A/p∶ A Ï” RA; 1 ≀ p2 = p Ï” A} is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate S(dRA) to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies 0 ≀ 1, and we prove that the identity 1 ≀ 0 holds only in trivial members of S(dRA).https://digitalcommons.chapman.edu/scs_books/1050/thumbnail.jp

    A Categorical Equivalence Motivated by Kalman’s Construction

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    An equivalence between the category of MV-algebras and the category MV∙ is given in Castiglioni et al. (Studia Logica 102(1):67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations a=¬¬a,(a→b)∹(b→a)=1 and a⊙(a→b)=a∧b. An object of MV∙ is a residuated lattice which in particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs (A, I), where A is an MV-algebra and I is an ideal of A.Facultad de Ciencias Exacta

    Quasi-Nelson Algebras

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    Abstract We introduce a generalization of Nelson algebras having a not-necessarily involutive negation; we suggest to dub this class quasi-Nelson algebras in analogy with quasi-De Morgan lattices, these being a non-involutive generalization of De Morgan lattices. We show that, similarly to the involutive case (and perhaps surprisingly), our new class of algebras can be equivalently presented as (1) quasi-Nelson residuated lattices, i.e. models of the well-known Full Lambek calculus with exchange and weakening, extended with the Nelson axiom; (2) non-involutive twist-structures, i.e. special products of Heyting algebras, which generalize the well-known construction for representing algebraic models of Nelson's constructive logic with strong negation; (3) quasi-Nelson algebras, i.e. models of non-involutive Nelson logic viewed as a conservative expansion of the negation-free fragment of intuitionistic logic. The equivalence of the three presentations, and in particular the extension of the twist-structure representation to the non-involutive case, is the main technical result of the paper. We hope, however, that the main impact may be the possibility of opening new ways to (i) obtain deeper insights into the distinguishing feature of Nelson's logic (the Nelson axiom) and its algebraic counterpart; (ii) be able to investigate certain purely algebraic properties (such as 3-potency and (0,1)-congruence orderability) in a more general setting
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