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EnglishVarieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a ô-regular variety V the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the ô-assertional logic of V. Moreover, if V has a constant 1 in its type and is 1-subtractive, the deductive filters on A 2 V of the 1-assertional logic of V coincide with the V-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and ô-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics. §
| Titolo: | Quasi-subtractive varieties |
| Autori: | |
| Data di pubblicazione: | 2011 |
| Rivista: | |
| Abstract: | Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a ô-regular variety V the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the ô-assertional logic of V. Moreover, if V has a constant 1 in its type and is 1-subtractive, the deductive filters on A 2 V of the 1-assertional logic of V coincide with the V-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and ô-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics. § |
| Handle: | http://hdl.handle.net/11584/28086 |
| Tipologia: | 1.1 Articolo in rivista |
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http://hdl.handle.net/11584/28086
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