(∈,∈∨q)-OSMOTIC VALUES WITH APPLICATIONS IN BCK/BCI-ALGEBRAS

Abstract

The notions of (\({∈}, {∈} ∨ {q}\))-osmotic S-value-osmotic S-value and (\(\overline{∈}, \overline{∈} ∨ \: \overline{q}\))-osmotic S-value are introduced, and related properties are investigated.  Relations between osmotic S-value and (\({∈}, {∈} ∨ {q}\))-osmotic S-value ((\(\overline{∈}, overline{∈} ∨ overline{q}\))-osmotic S-value) are considered.  Conditions for a number in the unit interval [0, 1] to be an osmotic S-value (resp., (\({∈}, {∈} ∨ {q}\))-osmotic S-value and (\(\overline{∈}, \overline{∈} ∨ \: \overline{q}\))-osmotic S-value) are provided.  Conditions for level sets to be S-energetic set and/or subalgebra are discussed.