Logic, Semantics, Metamathematics. By A. Tarski. Translated by J. H. Woodger. Pp. 471. 60s. 1956. (Clarendon Press)

Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.

A Survey of Residuated Lattices

A hoop is a naturally ordered pocrim (ie, a partially ordered commutative residuated integral monoid) We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its finite members

On the structure of hoops

Varieties of BL- algebras I: General properties.

Hajek acknowledges partial support by the grant No.A1030004/00 of the Grant Agency of the Academy of Sciences of the Czech Republic. Esteva and Godo acknowledge partial support by the Spanish CICYT project LOFAG TIC2001-1577-C03-01.

Hoops and Fuzzy Logic

A continuous t-norm is a continuous map * from [0, 1]2 into [0, 1] such that $$\langle [0, 1], *, 1 \rangle$$
 is a commutative totally ordered monoid. Since the natural ordering on [0, 1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and $$\langle [0, 1], *,\rightarrow, 1\rangle$$
 becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras $$\langle [0, 1], *,\rightarrow, 1\rangle$$
 , where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hajek’s BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [0, 1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL-algebras, of Godel algebras and of product algebras.

Basic Hoops: an Algebraic Study of Continuous t-norms

On varieties and quasivarieties of hoops and their reducts.

In this paper we review a hidden (sorted) generalization of k-deductive systems—hidden k-logics. They encompass deductive systems as well as hidden equational logics and inequational logics. The special case of hidden equational logics has been used to specify and to verify properties in program development of behavioral systems within the dichotomy visible vs. hidden data. We recall one of the main applications of this work—the study of behavioral equivalence. Related results are obtained through combinatorial properties of the Leibniz congruence relation.

/pdf/a-short-overview-of-hidden-logic-2mzni07bkv.pdf

Isabel M. A. Ferreirim

Papers

On the structure of hoops

Basic Hoops: an Algebraic Study of Continuous t-norms

On varieties and quasivarieties of hoops and their reducts.

A short overview of Hidden Logic