Basic Hoops: an Algebraic Study of Continuous t-norms

Abstract

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.

Free algebras in varieties of BL-algebras generated by a BLn-chain

“…A basic /loop [1] or a generalized BL-algebra [18], is a hoop that satisfies the equation It is shown in [1] that generalized BL-algebras can be characterized as algebras A = (A, A, v, *, -*, T) of type ( 2 , 2 , 2 , 2 , 0 ) such that (1) (A, *, T), is an commutative monoid, (2) L(A) := (A, A, v, T), is a lattice with greatest element T, (3) x -x = T, (4) x -»• (y -> z) = (JC * y) -» z, (5) j[A)i = jt*(j;->y), (6) U^y)v(y->;t) = T.…”

“…BL-algebras form a variety (or equational class) of residuated lattices [19]. More precisely, they can be characterized as bounded basic hoops [1,7]. Subvarieties of the variety of BL-algebras are in correspondence with axiomatic extensions of BL.…”

Multipotent GBL-algebras

“…In general, if X-algebras are defined and happen to form a variety, we will use sans-serif X for that variety. Thus, FL i is the variety of FL i -algebras: these are FL-algebras satisfying (1); they are also known as integral. FL-algebras satisfying (2) are called FL o -algebras, or zero-bounded.…”

“…Thus, an element a is n-potent if a n+1 = a n , and it is idempotent if it is 1-potent. For commutativity (10), we only remark that commutative integral GBL-algebras, also known as basic hoops (cf., e.g., [1,2]), are term equivalent to 0-free subreducts of BL-algebras, and commutative pseudo BL-algebras are BL-algebras. Similarly, commutative integral GMV-algebras, also known as Wajsberg hoops, are term equivalent to 0-free subreducts of MV-algebras, and commutative pseudo MV-algebras are MV-algebras.…”

Decomposition of BL-chains

“…Considering the fact that BL-algebras have as an algebraic root the theory of hoops (see [1]), Aglianò and Montagna give in [2] a theorem of decomposition for BL-chains (i.e., basic totally ordered bounded hoops) into some special kind of hoops, named Wajsberg hoops, which cannot be further decomposed. Although this improves the result given in [5] (for it does not need to introduce the notion of saturated BL-chain), the given proof strongly relies on the axiom of choice (as a matter of fact, it is invoked three times in the course of the proof).…”