INT-SOFT SEMIGROUPS WITH TWO THRESHOLDS

In Suk Kong
2016 Honam Mathematical Journal  
In this paper, we study more general version of the paper [J. H. Lee, I. S. Kong, H. S. Kim and J. U. Jung, Generalized int-soft subsemigroups, Ann. Fuzzy Math. Inform. 8(6) (2014) 869-887]. We introduce the notion of int-soft semigroup with two thresholds ε and δ (briefly, (ε, δ)-int-soft semigroup) of a semigroup S, and investigate several related properties. words and phrases. int-soft semigroup, θ-generalized int-soft semigroup, (ε, δ)-int-soft semigroup. 96 I. S. Kong discussed the soft
more » ... -image and soft image of a θ-generalized int-soft subsemigroup under the homomorphism. The aim of this paper is to study more general version of the paper [15] . We introduce the notion of int-soft semigroup with two thresholds ε and δ (briefly, (ε, δ)-int-soft semigroup) of a semigroup S, and provide many examples. We investigate the following items: 1. Relations between a δ-generalized int-soft semigroup over U and an (ε, δ)-int-soft semigroup over U . 2. Relations between an ε-generalized int-soft semigroup over U and an (ε, δ)-int-soft semigroup over U . 3. Relations between a δ-generalized int-soft semigroup over U and an ε ∩ δ-generalized int-soft semigroup over U . 4. Relations between an ε-generalized int-soft semigroup over U and an ε ∩ δ-generalized int-soft semigroup over U . Relations between an ε ∪ δ-generalized int-soft semigroup over U and an ε-generalized int-soft semigroup over U . 6. Relations between an ε ∪ δ-generalized int-soft semigroup over U and δ-generalized int-soft semigroup over U . 7. Relations between an ε ∪ δ-generalized int-soft semigroup over U and an ε ∩ δ-generalized int-soft semigroup over U . 8. Relations between an ε ∪ δ-generalized int-soft semigroup over U and an (ε, δ)-int-soft semigroup over U . 9. Relations between an ε ∩ δ-generalized int-soft semigroup over U and an (ε, δ)-int-soft semigroup over U . 10. Relations between a δ-generalized int-soft semigroup over U and an ε-generalized int-soft semigroup over U . Preliminaries Let S be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows: Molodtsov [18] defined the soft set in the following way: Let U be an initial universe set and E be a set of parameters. Let P(U ) denotes the power set of U and A ⊂ E.
doi:10.5831/hmj.2016.38.1.95 fatcat:gjb3v25z7fdy5po7aafxtfwlla