Lattice-valued soft algebras

Motivated by the rapidly developing theory of lattice-valued (or, more generally, variety-based) topological systems, which takes its origin in the crisp concept of S. Vickers (introduced as a common framework for both topological spaces and their underlying algebraic structures—frames or locales), the paper initiates a deeper study of one of its incorporated mathematical machineries, i.e., the realm of soft sets of D. Molodtsov. More precisely, we start the theory of lattice-valued soft universal algebra, which is based in soft sets and lattice-valued algebras of A. Di Nola and G. Gerla. In particular, we provide a procedure for obtaining soft versions of algebraic structures and their homomorphisms, as well as basic tools for their investigation. The proposed machinery underlies many concepts of (lattice-valued) soft algebra, which are currently available in the literature, thereby enabling the respective researchers to avoid its reinvention in future.