Soft subsets and soft product operations

Molodtsov’s soft set theory provides a general mathematical framework for dealing with uncertainty. It is known that soft subsets and soft equal relations are of vital importance in soft set theory. This paper aims to give a systematic study on several types of soft subsets and various soft equal relations induced by them. We give some equivalent characterizations of different soft subsets and endeavor to ascertain the interrelations among these notions, illustrated by a number of concrete examples. We also consider ontology-based soft sets and show that soft L-subsets generalize both soft M-subsets and ontology-based soft subsets. Moreover, by means of soft L-subsets and some related notions, we give a theoretical study concerning soft product operations such as ∧-products and ∨-products. We consummate some incomplete results concerning soft product operations existing in the literature, and investigate the algebraic properties of soft product operations in detail. Finally, we consider free soft algebras associated with soft product operations. It is shown that soft L-equal relations are congruence relations over free soft algebras and the corresponding quotient structures form commutative semigroups.