Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines

Fuzzy Sets and Systems 20 (1986) 107-108 North-Holland

107

ERRATA A categorical

accommodation

of various notions of fuzzy topology

Fuzzy Sets and Systems 9 (1983) 241-265

Complete fuzzy topological hyperfields and fuzzy multiplication fuzzy real lines Fuzzy Sets and Systems 15 (1985) 285-310

in the

S.E. RODABAUGH Department Youngstown,

of

Mathematical OH 44555,

and

Compurer

Sciences,

Youngstown

State

University,

USA

Received April 1986

The purpose of this note is to correct a mistake made in [I] and carried over into [2]: contrary to what is claimed in [l, 21, %(L, 9) of [2] (%(L, r$) of [l]) is not necessarily a category, and hence not necessarily a subcategory of T [2] (or FUZZ ill); U(L, 9) need only be a subcollection of objects and morphisms of U. It is a subcategory of U if $ = idL. To state precisely the corrigenda for [l, 21, we define the category U.+,(L,@) introduced in [3]: (X, L, t) E U,(L, Cp) iff (X, L, t) E U and (Vb E Lx, b E t iff $-lob E ~1, and cf, #)o(g, $)= (fog, $1. N ow U,(L, @) is a category for each 9-l E Endo( but need not be a subcategory of U. This is easily remedied by invoking the categorical isomorphism F of [3] taking U,(L, @) onto a full subcategeory of U(L, id,-) via F(X, L, r) = (X L, ~1,

W, $I=

(f, idA

and so for each L, for each @-’ E Endo( U,(L, $) may be regarded as a subcategory of U. We also need the adjunction F -I G+ from [3] to which we refer the reader for details and proofs of the foregoing and related ideas. Using the preceding paragraph, the corrigenda for [l, 21 can be stated precisely: for [l], either replace ‘[quasi-full, full] subcategory’ with ‘[quasi-full, full] subcollection of objects and morphisms’ in Definition 3.4, Definition 3.5, Convention 4.1, and the paragraph preceding Theorem 4.1, or in these results replace %(L, @) [%A& $1, Se,(L, $1, ‘;e,(L, $1, %(L, $11 with T+(L, $1

[U&L, cp),

U&L,

#),

U,,(L, 4')) U,,(L, @)I; and in [2] either replace

‘functor’ in Proposition 2.1, Definition 5.1 with ‘functoriale’ [3] (a map on a subcollection of objects and morphisms which preserves horn-sets, i.e. it maps an arrow between two objects to an arrow of the same direction between images of the two objects), or in Proposition 2.1 replace U,(L, $) by U&L, @) and 0165-0114/86/$3.50 @ 1986, Elsevier Science Publishers B.V. (North-Holland)

108

Errata

stipulate C#JtzAuto(L), in Definition 5.1 replace T(L, C#J)by U,(L, $); furthermore, the @-product of [2] is not generally a categorical product - see Definition 2.2, Proposition 2.2 of [2] - but it becomes categorical if C#J= idL or T(L, $I) is replaced with lJ,(L, @) and (either CpE Auto(L) or the @-product is modified by the G+ functor to create the new &product of [3]).

References [l] SE. Rodabaugh, A categorical accommodation of various notions of fuzzy topology, Fuzzy Sets and Systems 9 (1983) 241-265. [2] S.E. Rodabaugh, Complete fuzzy topological hypertields and fuzzy multiplication in the fuzzy real lines, Fuzzy Sets and Systems 15 (1985) 285-310. [3] S.E. Rodabaugh, A point-set lattice-theoretic framework T for topology which contains LOC as a subcategory of singleton spaces and in which there are general classes of Stone representation and compactification theorems, Preprint (1986).