LEADER 05476nam 2200673Ia 450 001 9910784596803321 005 20200520144314.0 010 $a1-281-05106-3 010 $a9786611051068 010 $a0-08-048964-8 035 $a(CKB)1000000000357860 035 $a(EBL)294345 035 $a(OCoLC)469589702 035 $a(SSID)ssj0000236699 035 $a(PQKBManifestationID)11235163 035 $a(PQKBTitleCode)TC0000236699 035 $a(PQKBWorkID)10172941 035 $a(PQKB)11742018 035 $a(Au-PeEL)EBL294345 035 $a(CaPaEBR)ebr10186004 035 $a(CaONFJC)MIL105106 035 $a(MiAaPQ)EBC294345 035 $a(EXLCZ)991000000000357860 100 $a20070511d2007 uy 0 101 0 $aeng 135 $aurcn||||||||| 181 $ctxt 182 $cc 183 $acr 200 00$aResiduated lattices$b[electronic resource] $ean algebraic glimpse at substructural logics /$fNikolaos Galatos ... [et al.] 205 $a1st ed. 210 $aAmsterdam ;$aBoston $cElsevier$d2007 215 $a1 online resource (532 p.) 225 1 $aStudies in logic and the foundations of mathematics,$x0049-237X ;$vv. 151 300 $aDescription based upon print version of record. 311 $a0-444-52141-0 320 $aIncludes bibliographical references (p. 479-495) and index. 327 $aCover; Copyright Page; Table of Contents; Detailed Contents; List of Figures; List of Tables; Introduction; Chapter 1. Getting started; 1.1. First-order languages and semantics; 1.1.1. Preorders; 1.1.2. Posets; 1.1.3. Lattices; 1.1.4. Heyting algebras and Boolean algebras; 1.1.5. Semigroups, monoids and other groupoids; 1.2. Concepts from universal algebra; 1.2.1. Homomorphisms, subalgebras, substructures, direct products; 1.2.2. Congruences; 1.2.3. Free algebras; 1.2.4. More on Heyting and Boolean algebras; 1.2.5. Mal'cev conditions; 1.2.6. Ultraproducts and Jo?nsson's Lemma 327 $a1.2.7. Equational logic 1.2.8. Quasivarieties; 1.3. Logic; 1.3.1. Hilbert calculus for classical logic; 1.3.2. Gentzen's sequent calculus for classical logic; 1.3.3. Calculi for intuitionistic logic; 1.3.4. Provability in Hilbert and Gentzen calculi; 1.4. Logic and algebra; 1.4.1. Validity of formulas in algebras; 1.4.2. Lindenbaum-Tarski algebras; 1.4.3. Algebraization; 1.4.4. Superintuitionistic logics; 1.5. Cut elimination in sequent calculi; 1.5.1. Cut elimination; 1.5.2. Decidability and subformula property; 1.6. Consequence relations and matrices; 1.6.1. Consequence relations 327 $a1.6.2. Inference rules 1.6.3. Proofs and theorems; 1.6.4. Matrices; 1.6.5. Examples; 1.6.6. First-order and (quasi)equational logic; Exercises; Notes; Chapter 2. Substructural logics and residuated lattices; 2.1. Sequent calculi and substructural logics; 2.1.1. Structural rules; 2.1.2. Comma, fusion and implication; 2.1.3. Sequent calculus for the substructural logic FL; 2.1.4. Deducibility and substructural logics over FL; 2.2. Residuated lattices and FL-algebras; 2.3. Important subclasses of substructural logics; 2.3.1. Lambek calculus; 2.3.2. BCK logic and algebras; 2.3.3. Relevant logics 327 $a2.3.4. Linear logic 2.3.5. ukasiewicz logic and MV-algebras; 2.3.6. Fuzzy logics and triangular norms; 2.3.7. Superintuitionistic logics and Heyting algebras; 2.3.8. Minimal logic and Brouwerian algebras; 2.3.9. Fregean logics and equivalential algebras; 2.3.10. Overview of logics over FL; 2.4. Parametrized local deduction theorem; 2.5. Hilbert systems; 2.5.1. The systems HFLe and HFL; 2.5.2. Derivable rules; 2.5.3. Equality of two consequence relations; 2.6. Algebraization and deductive filters; 2.6.1. Algebraization; 2.6.2. Deductive filters; Exercises; Notes 327 $aChapter 3. Residuation and structure theory 3.1. Residuation theory and Galois connections; 3.1.1. Residuated pairs; 3.1.2. Galois connections; 3.1.3. Binary residuated maps; 3.2. Residuated structures; 3.3. Involutive residuated structures; 3.3.1. Involutive posets; 3.3.2. Involutive pogroupoids; 3.3.3. Involutive division posets; 3.3.4. Term equivalences; 3.3.5. Constants; 3.3.6. Dual algebras; 3.4. Further examples of residuated structures; 3.4.1. Boolean algebras and generalized Boolean algebras; 3.4.2. Partially ordered and lattice ordered groups 327 $a3.4.3. The negative cone of a residuated lattice 330 $aThe book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as 410 0$aStudies in logic and the foundations of mathematics ;$vv. 151. 606 $aAlgebraic logic 606 $aLattice theory 615 0$aAlgebraic logic. 615 0$aLattice theory. 676 $a511.33 686 $a31.10$2bcl 701 $aGalatos$b Nikolaos$0731529 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910784596803321 996 $aResiduated lattices$91441196 997 $aUNINA