An extension of the fuzzy unit interval to a tensor product with completely distributive first factor

Abstract

The original Hutton interval I (L) can algebraically be identified with the tensor product I ⊗ L of the real unit interval I and a complete lattice L. Due to this, the tensor product M ⊗ L with M a completely distributive lattice is considered as a generalization of the lattice I (L). When appropriately endowed with an L-topology, the tensor product M ⊗ L becomes also an L-topological extension of I (L). If M is ▹-separable (= it has a countable join base free of supercompact elements), many of the L-topological features of I (L) are retained. To wit, Urysohn lemma and Tietze–Urysohn extension theorem for (M ⊗ L)-valued functions are then proved. The relationship of M ⊗ L to the L-fuzzy topological modification of M in the sense of D. Zhang and Y.-M. Liu [27] is discussed.