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The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notions of closed intersectional soft BCI-ideals and intersectional soft commutative BCI-ideals are introduced, and related properties are investigated. Conditions for an intersectional soft BCI-ideal to be closed are provided. Characterizations of an intersectional soft commutative BCI-ideal are established, and a new intersectional soft c-BCI-ideal from an old one is constructed. [ABSTRACT FROM AUTHOR]

The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a quasi (or, pseudo) star-shaped set to be a star-shaped set are provided. [ABSTRACT FROM AUTHOR]

Intuitionistic falling shadow is introduced, and applied to B C K / B C I -algebras. Falling intuitionistic subalgebra and falling intuitionistic ideal of B C K / B C I -algebras are introduced, and related properties are investigated. Relations between falling intuitionistic subalgebra and falling intuitionistic ideal are discussed. A characterization of falling intuitionistic ideal is established. [ABSTRACT FROM AUTHOR]

The notions of int-soft filters, int-soft G-filters, regular int-soft filters, and MV-int-soft filters in residuated lattices are introduced, and their relations, properties, and characterizations are investigated. Conditions for an int-soft filter to be an int-soft G-filter, a regular int-soft filter, or an MV-int-soft filter are provided. The extension property for an int-soft G-filter is discussed. Finally, it is shown that the notion of an MV-int-soft filter coincides with the notion of a regular int-soft filter in BL-algebras.