An extension of the fuzzy unit interval to a tensor product with completely distributive first factor ☆ ☆The authors gratefully acknowledge support from the Ministry of Economy and Competitiveness of Spain (grant MTM2015-63608-P (MINECO/FEDER)). The first named author also acknowledges support from the Basque Government (grant IT974-16).

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.