id: 05772813
dt: a
an: 05772813
au: Metcalfe, George; Paoli, Francesco; Tsinakis, Constantine
ti: Ordered algebras and logic.
so: Hosni, Hykel (ed.) et al., Probability, uncertainty and rationality. Pisa:
    Edizioni della Normale (ISBN 978-88-7642-347-5/hbk). Centro di Ricerca
    Matematica Ennio De Giorgi (CRM) Series (Nuova Serie) 10, 3-83 (2010).
py: 2010
pu: Pisa: Edizioni della Normale
la: EN
cc: 
ut: substructural logics; residuated lattices; amalgamation; interpolation;
    substructural proof theory; deductive filters; ideal theory of rings;
    Boolean algebras; Heyting algebras; survey paper
ci: 
li: 
ab: The main scope of this survey paper is to focus on the relationships
    between substructural logics and residuated lattices.  On the one hand,
    substructural logics encompass many important nonclassical logics, such
    as the full Lambek calculus, linear logic, relevance logics, and fuzzy
    logics.  On the other hand, residuated lattices, while providing
    algebraic semantics for these logics, also feature in areas interesting
    from the order-algebraic perspective, such as the theory of
    lattice-ordered groups, vector lattices (or Riesz spaces) and abstract
    ideal theory. Algebraic methods have been used to address completeness
    problems for Gentzen systems, while these systems have been used to
    establish decidability and amalgamation properties for classes of
    algebras.  The first aim of this article is to briefly trace the
    distinct historical roots of ordered algebras and logic. The second aim
    is to explain and illustrate the usefulness of this theory for ordered
    algebra and logic in the context of residuated lattices and
    substructural logics. In particular the authors explain how completions
    on the ordered algebra side, and Gentzen systems on the logic side, are
    used to address properties such as decidability, interpolation and
    amalgamation, and completeness.
rv: Dana Piciu (Craiova)