Variable-basis Categorically-algebraic Dualities

The manuscript continues our study on developing a categorically-algebraic (catalg) analogue of the theory of natural dualities of D. Clark and B. Davey, which provides a machinery for obtaining topological representations of algebraic structures. The new setting differs from its predecessor in relying on catalg topology, introduced lately by the author as a new approach to topological structures, which incorporates the majority of both crisp and many-valued developments, ultimately erasing the border between them. Motivated by the variable-basis lattice-valued extension of the Stone representation theorems done by S. E. Rodabaugh, we have recently presented a catalg version of the Priestley duality for distributive lattices, which gave rise (as in the classical case) to a fixed-basis variety-based approach to natural dualities. In this paper, we extend the theory to variable-basis, whose setting is completely different from the respective one of S. E. Rodabaugh, restricted to isomorphisms between the underlying lattices of the spaces.