275 research outputs found
Canonical formulas for k-potent commutative, integral, residuated lattices
Canonical formulas are a powerful tool for studying intuitionistic and modal
logics. Actually, they provide a uniform and semantic way to axiomatise all
extensions of intuitionistic logic and all modal logics above K4. Although the
method originally hinged on the relational semantics of those logics, recently
it has been completely recast in algebraic terms. In this new perspective
canonical formulas are built from a finite subdirectly irreducible algebra by
describing completely the behaviour of some operations and only partially the
behaviour of some others. In this paper we export the machinery of canonical
formulas to substructural logics by introducing canonical formulas for
-potent, commutative, integral, residuated lattices (-).
We show that any subvariety of - is axiomatised by canonical
formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde
Distributive Residuated Frames and Generalized Bunched Implication Algebras
We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames
The ubiquity of conservative translations
We study the notion of conservative translation between logics introduced by
Feitosa and D'Ottaviano. We show that classical propositional logic (CPC) is
universal in the sense that every finitary consequence relation over a
countable set of formulas can be conservatively translated into CPC. The
translation is computable if the consequence relation is decidable. More
generally, we show that one can take instead of CPC a broad class of logics
(extensions of a certain fragment of full Lambek calculus FL) including most
nonclassical logics studied in the literature, hence in a sense, (almost) any
two reasonable deductive systems can be conservatively translated into each
other. We also provide some counterexamples, in particular the paraconsistent
logic LP is not universal.Comment: 15 pages; to appear in Review of Symbolic Logi
Gluing residuated lattices
We introduce and characterize various gluing constructions for residuated
lattices that intersect on a common subreduct, and which are subalgebras, or
appropriate subreducts, of the resulting structure. Starting from the 1-sum
construction (also known as ordinal sum for residuated structures), where
algebras that intersect only in the top element are glued together, we first
consider the gluing on a congruence filter, and then add a lattice ideal as
well. We characterize such constructions in terms of (possibly partial)
operators acting on (possibly partial) residuated structures. As particular
examples of gluing constructions, we obtain the non-commutative version of some
rotation constructions, and an interesting variety of semilinear residuated
lattices that are 2-potent. This study also serves as a first attempt toward
the study of amalgamation of non-commutative residuated lattices, by
constructing an amalgam in the special case where the common subalgebra in the
V-formation is either a special (congruence) filter or the union of a filter
and an ideal.Comment: This is a preprint. The final version of this work appears in Orde
Weakening Relation Algebras and FL\u3csup\u3e2\u3c/sup\u3e-algebras
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/pⶠA ϔ RA; 1 †p2 = p ϔ 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
The Structure of Generalized BI-algebras and Weakening Relation Algebras
Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under GummâUrsini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comerâs double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras
Most simple extensions of are undecidable
All known structural extensions of the substructural logic ,
Full Lambek calculus with exchange/commutativity, (corresponding to
subvarieties of commutative residuated lattices axiomatized by -equations) have decidable theoremhood; in particular all the ones defined
by knotted axioms enjoy strong decidability properties (such as the finite
embeddability property). We provide infinitely many such extensions that have
undecidable theoremhood, by encoding machines with undecidable halting problem.
An even bigger class of extensions is shown to have undecidable deducibility
problem (the corresponding varieties of residuated lattices have undecidable
word problem); actually with very few exceptions, such as the knotted axioms
and the other prespinal axioms, we prove that undecidability is ubiquitous.
Known undecidability results for non-commutative extensions use an encoding
that fails in the presence of commutativity, so and-branching counter machines
are employed. Even these machines provide encodings that fail to capture proper
extensions of commutativity, therefore we introduce a new variant that works on
an exponential scale. The correctness of the encoding is established by
employing the theory of residuated frames.Comment: 45 page
Theorems of Alternatives for Substructural Logics
A theorem of alternatives provides a reduction of validity in a substructural
logic to validity in its multiplicative fragment. Notable examples include a
theorem of Arnon Avron that reduces the validity of a disjunction of
multiplicative formulas in the R-mingle logic RM to the validity of a linear
combination of these formulas, and Gordan's theorem for solutions of linear
systems over the real numbers, that yields an analogous reduction for validity
in Abelian logic A. In this paper, general conditions are provided for
axiomatic extensions of involutive uninorm logic without additive constants to
admit a theorem of alternatives. It is also shown that a theorem of
alternatives for a logic can be used to establish (uniform) deductive
interpolation and completeness with respect to a class of dense totally ordered
residuated lattices
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