By Stephen Ernest Rodabaugh, Erich Peter Klement, Ulrich Höhle (auth.), Stephen Ernest Rodabaugh, Erich Peter Klement, Ulrich Höhle (eds.)

This publication has a primary dating to the foreign Seminar on Fuzzy Set concept held each one September in Linz, Austria. First, this quantity is a longer account of the 11th Seminar of 1989. moment, and extra importantly, it's the end result of the culture of the previous ten Seminars. the aim of the Linz Seminar, when you consider that its inception, used to be and is to foster the advance of the mathematical points of fuzzy units. within the prior years, this was once comprehensive by way of bringing jointly for per week small grou playstation of mathematicians in quite a few fields in an intimate, concentrated setting which promoted a lot casual, serious dialogue as well as formal displays. starting with the 10th Seminar, the intimate surroundings was once retained, yet every one Seminar narrowed in subject matter; and participation used to be broadened to incorporate either more youthful students inside of, and verified mathematicians outdoor, the mathematical mainstream of fuzzy units thought. lots of the fabric of this booklet used to be constructed through the years in shut organization with the Seminar or motivated by means of what transpired at Linz. for a lot of the content material, it performed an important position in both stimulating this fabric or in supplying suggestions and the mandatory screening of principles. hence we might relatively say that the booklet, and the 11th Seminar to which it really is at once similar, are in lots of respects a fruits of the former Seminars.

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MacNeille 1937]) is a GL-monoid satisfying also (L6). For each subset A of P let us denote by U(A) (resp. L(A)) the set of all upper (resp. lower) bounds of A; then the MacNeille completion (P#, :::;#) of (P,:::;) is determined by PROOF. p# = {A ~ P I L(U(A)) = A} A :::;# B The semigroup operation multiplication A * B = {a (a) We show {::> A ~ B. * induces on the power set * b I a E A, bE B}. 3) The set inclusion U[L(U(A)) * B] ~ U(A * B) is obvious. On the other hand we fix E L(U(A)), bo E B and choose i E U(A * B).

0 The MacNeille completion preserves also the axioms (L3) and (L5). With respect to later applications we consider explicitly the case of MV-algebras. 6'. e. an integral, residuated, commutative po-monoid with zero satisfying (LO) and (L5). If (P, $, *, -» fulfills (L6), then the MacNeille completion (P#, $#, *#) is a complete MV-algebra. PROOF. 4)) is a subset of the residual of A by B in (P#, $#, *#). Obviously B ! 0# coincides with L({b -+ 0 I bE B}) =: D. 7) If dE D, then d :5 b -> 0 for all bE B; hence d -> 0 is an upper bound of B.

4. By addition of further axioms, the weak subobject classifier axiom can be enriched in such a way which leads to the concept of weak toposes (cf. [Hohle 1990b], [Hohle and Stout 1991]). In this context it is interesting to note that toposes and finitely cocomplete collapsed toposes are also weak toposes. Further we emphasize that weak toposes have an internal logic, which coincides in the case of CMETI (= category of complete metric spaces with non-expansive maps) with the [0, i]-valued Lukasiewicz logic (cf.