scholarly journals Proof theory for lattice-ordered groups

10.29007/3szk ◽  
2018 ◽  
Author(s):  
George Metcalfe
Keyword(s):  
Proof Theory ◽  
Family Resemblance ◽  
Substructural Logics ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Theoretic Approach ◽  
Gentzen Systems ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebras

Proof theory can provide useful tools for tackling problems in algebra. In particular, Gentzen systems admitting cut-eliminationhave been used to establish decidability, complexity, amalgamation, admissibility, and generation results for varieties of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing some family resemblance to groups, such as lattice-ordered groups, MV-algebras, BL-algebras, and cancellative residuated lattices, the proof-theoretic approach has met so far only with limited success.The main aim of this talk will be to introduce proof-theoretic methods for the class of lattice-ordered groups and to explain how these methods can be used to obtain new syntactic proofs of two core theorems: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

Substructural fuzzy logics

2007 ◽  
Vol 72 (3) ◽  
pp. 834-864 ◽  
Author(s):  
George Metcalfe ◽  
Franco Montagna
Keyword(s):  
Linear Logic ◽  
Substructural Logics ◽  
Unit Interval ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
The Real ◽  
Gentzen Systems ◽  
Intuitionistic Linear Logic

AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

scholarly journals The Riesz Hull of a Semisimple MV-Algebra

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
D. Diaconescu ◽  
I. Leuștean
Keyword(s):  
Vector Lattice ◽  
Strong Unit ◽  
Riesz Spaces ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Standard Construction ◽  
Abelian Lattice ◽  
Mv Algebras

AbstractMV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

scholarly journals Equivalence à la Mundici for commutative lattice-ordered monoids

2021 ◽  
Vol 82 (3) ◽  
Author(s):  
Marco Abbadini
Keyword(s):  
Unary Operation ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Abelian Lattice ◽  
Mv Algebras

AbstractWe provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation $$x \mapsto -x$$ x ↦ - x . The primitive operations are $$+$$ + , $$\vee $$ ∨ , $$\wedge $$ ∧ , 0, 1, $$-1$$ - 1 . A prime example of these structures is $$\mathbb {R}$$ R , with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $$x \mapsto \lnot x$$ x ↦ ¬ x . The primitive operations are $$\oplus $$ ⊕ , $$\odot $$ ⊙ , $$\vee $$ ∨ , $$\wedge $$ ∧ , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $$[0, 1]\subseteq \mathbb {R}$$ [ 0 , 1 ] ⊆ R . We obtain the original Mundici’s equivalence as a corollary of our main result.

A relationship between the category of chain MV-algebras and a subcategory of abelian groups

2021 ◽  
Vol 71 (4) ◽  
pp. 1027-1045
Author(s):  
Homeira Pajoohesh
Keyword(s):  
Abelian Groups ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice ◽  
Mv Algebras

Abstract The category of MV-algebras is equivalent to the category of abelian lattice ordered groups with strong units. In this article we introduce the category of circled abelian groups and prove that the category of chain MV-algebras is isomorphic with the category of chain circled abelian groups. In the last section we show that the category of chain MV-algebras is a subcategory of abelian cyclically ordered groups.

Banaschewski’s theorem for GMV-algebras

2012 ◽  
Vol 62 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Subdirect Product ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebras

AbstractBanaschewski’s theorem concerns subdirect product decompositions of lattice ordered groups. In the present paper we deal with the analogous investigation for the case of generalized MV -algebras (GMV -algebras, in short); we apply this notion in the sense studied by Galatos and Tsinakis.

Weak relatively uniform convergences on MV-algebras

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Štefan Černák ◽  
Ján Jakubík
Keyword(s):  
Present Article ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Brouwerian Lattice ◽  
Mv Algebras ◽  
Natural Way

AbstractWeak relatively uniform convergences (wru-convergences, for short) in lattice ordered groups have been investigated in previous authors’ papers. In the present article, the analogous notion for MV-algebras is studied. The system s(A) of all wru-convergences on an MV-algebra A is considered; this system is partially ordered in a natural way. Assuming that the MV-algebra A is divisible, we prove that s(A) is a Brouwerian lattice and that there exists an isomorphism of s(A) into the system s(G) of all wru-convergences on the lattice ordered group G corresponding to the MV-algebra A. Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A).

scholarly journals Proof theory for lattice-ordered groups

2016 ◽  
Vol 167 (8) ◽  
pp. 707-724 ◽  
Author(s):  
Nikolaos Galatos ◽  
George Metcalfe
Keyword(s):  
Proof Theory ◽  
Ordered Groups ◽  
Lattice Ordered Groups

scholarly journals Algebraic proof theory for substructural logics: Cut-elimination and completions

2012 ◽  
Vol 163 (3) ◽  
pp. 266-290 ◽  
Author(s):  
Agata Ciabattoni ◽  
Nikolaos Galatos ◽  
Kazushige Terui
Keyword(s):  
Proof Theory ◽  
Substructural Logics ◽  
Cut Elimination ◽  
Algebraic Proof
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