A note on tensor products on the unit interval. (English) Zbl 0358.18007

Menu, Jan; Pavelka, Jan

Commentat. Math. Univ. Carol. 17, 71-83 (1976).

Under the usual definition of the complement of a fuzzy subset \(A\): \(U \to [0,1]\) of a universe \(U\) by the formula \((\sim A)(x) = 1 - A(x)\), the complement \(\sim A\) fails to be a pseudocomplement in the lattice of fuzzy subsets of \(U\). To remedy this and related failings, the authors study (symmetric monoidal) closedness structures on the unit interval \(I\) viewed as a category in its usual ordering. A closedness structure on \(I\) with unit \(1\) is determined by a pair \((\square,h)\) where \(\square\): \(I \times I \to I\) (the tensor product) is an isotone map such that \((I,\square,1)\) is a commutative monoid and \(h: I^{OP} \times I \to I\) (the hom-product) is an isotone map such that \(x\square y \leq z\) iff \(x \leq h(y,z)\). Then \(\square\) is lower semicontinuous and \(h\) is upper semicontinuous for the usual topology on \(I \times I\). Using results on topological semigroups of i.a. P. S.Mostert and A. L. Shields [Ann. of Math., II. Ser. 65, 117–143 (1957; Zbl 0096.01203)] the authors give some examples of closedness structures on \(I\) and prove: (1) If \(h\) is continuous then \(\square\) is continuous and equivalent to the operation \(\max\{0,x+y-1\}\); (2) If \(h\) is continuous except at \((0,0)\), then \(\square\) is continuous and equivalent to the usual multiplication of real numbers; (3) The discontinuity set of \(h\) alone does not in general determine whether \(\square\) is continuous.

Reviewer: G. C. L. Brümmer

MSC:

18D15	Closed categories (closed monoidal and Cartesian closed categories, etc.)
22A15	Structure of topological semigroups
06F05	Ordered semigroups and monoids

Citations:

Zbl 0096.01203