A hoop is a naturally ordered pocrim (i.e., 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.

A continuous t-norm is a continuous map * from [0, 1] 2 into [0, 1] such that [0, 1], * , 1 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 [0, 1], * , →, 1 becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras [0, 1], * , →, 1 , 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 Hájek'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 Gödel algebras and of product algebras.A continuous t-norm is a continuous map * from [0, 1] 2 into [0, 1] such that [0, 1], * , 1 is a commutative totally ordered monoid. There are three fundamental continuous t-norms: the Lukasiewicz t-norm defined by x * L y = max(x + y − 1, 0), the Gödel (or lattice) norm x * G y = x ∧ y and the product norm x * P y = xy. Indeed it is known ([24, 35]) that, up to isomorphism, every continuous t-norm behaves locally as one of the above.Since the natural ordering on [0, 1] is a complete lattice ordering, each t-norm induces naturally a residuation, or an implication in more logical terms, by x → y = sup{z : z * x ≤ y}. The implications associated to the three fundamental norms are:x → L y = min(y − x + 1, 1)x → G y = 1 if x ≤ y y otherwiseResearch partly supported by research projects Praxis 2/ 2.

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In this paper we review a hidden (sorted) generalization of kdeductive 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.In addition we obtain a few new developments concerning hidden equational logic, namely we present a new characterization of the behavioral consequences of a theory.