$VOÖKDQ6H]JLQ 1 , * , $OH\QDùOJLQ 2 1 Department of Mathematics and Science Education, Faculty of Education, Amasya University, Amasya, Türkiye ; aslihan.sezgin@amasya.edu.tr . 2 Department of Mathematics, Graduate School of Natural and Applied Sciences, Amasya University, Amasya, Türkiye ; aleynailgiin@gmail.com . Received: 24 July 2023 Sezgin , A., & G•’— , A . ( 2024 ). - Revised: 09 October 2023 .

Soft computing fusion with applications , 1 ( 1 ), 2 8 - 4 3 . Accepted: 16 January 2024

The concept of bi - quasi - ideal generalizes the notions of bi - ideals and quasi - ideals in a semigroup; similarly, the soft intersection bi - quasi - ideal generalizes the concepts of soft intersection bi - ideals and soft intersection quasi - ideals in a semigroup. I n this paper, we introduce the concept of soft intersection almost bi - quasi ideal and its generalized concept, soft intersection weakly almost bi - quasi ideals, in a semigroup. In contrast to the soft intersection ideal theory, we demonstrate that every sof - quasi ideal is also a soft intersection almost ideal and a - ideal. Additionally, we show that every idempotent soft intersection almost bi - is a soft intersection almost subsemigroup, a soft in tersection almost weak interior ideal, a soft intersection almost tri - ideal, and a soft intersection almost tri - bi - ideal. Furthermore, we derive several interesting relationships regarding minimality, primeness, semiprimeness, and strong primeness between - quasi ideals and soft intersection - quasi ideals with the proven theorem stating that if a nonempty set is an almost bi - quasi ideal, then its soft characteristic function is also a soft intersection almost bi - quasi ideal, and vice versa .

Soft set, Bi - quasi ideals, Soft intersection ( Almost ) bi - qu .

Semigroups are crucial in many branches of mathematics because they provide the abstract algebraic foundation for memoryless systems, which restart on each iteration. Semigroups were first explored formally in the early 1900s. Semigroups are fundamental mo dels for linear time - invariant systems in practical mathematics. Studying finite semigroups is crucial to theoretical computer science, as they are naturally related to finite automata. In addition, semigroups and Markov processes are connected in probabil ity theory .

Ideals are required to comprehend algebraic structures and their uses. In 1952, bi - ideals for semigroups were initially presented by Good and Hughes [1] . The concept of quasi - ideals was introduced by Steinfeld [2] initially for semigroups and then extended to rings. Generalizing ideals in algebraic structures has been a major study area for many mathematicians. Corresponding Author: aslihan.sezgin@amasya.edu.tr jfea @aihe.ac.ir https://doi.org/10.22105/scfa.v1i1.26 Licensee System Analytics . This article is an open - access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0).

Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3 In 1980, Grosek and Satko [3] originally introduced the idea of almost left, right, and two - sided ideals of semigroups. Bogdanovic [4] generalized the notion of bi - ideals to almost bi - ideals in semigroups later, in 1981. In 2018, Wattanatripop et al. [5] defined almost quasi - ideals by combining the ideas of almost ideals and quasi - ideals of semigroups, expanding on the concepts of almost ideals and interior ideals of semigroups, and examining their characteristics. Kaopusek et al .

[6] proposed almost interior ideals and weakly almost interior ideals of semigroups in 2020. Iampan et al. [7] in 202 1 , Chinram and Nakkhasen [8] in 2022, Gaketem [9] in 2022, and Gaketem and Chinram [10] in 2023, Resp ectively, subsequently introduced almost subsemigroups, - quasi - interior ideals, almost bi - interior ideals, and almost bi - Additionally, several almost fuzzy semigroup ideal types were investigated in [5] , [7 ] ² [ 12] .

In 1999, Molodtsov [13] was the first to propose the idea of the soft set as a way to model uncertainty; this idea has subsequently drawn attention from a variety of fields. In [14 ] ² [ 23] , the fundamental operations of soft sets were examined. dDøPDQDQG(QJLQRøOX [24] modified the idea and presented soft intersection groups [25] , which sparked studies on a number of soft algebraic systems. As thoroughly reviewed in [26] , [27] , soft sets were also conveyed to semigroup theory with the concepts of semigroups with soft intersection left, right, and two - sided ideals, quasi - ideals, interior ideals, and generalized bi - ideals. Different semigroups were categorized by Sezgin and Orbay [28] using soft intersection substructures.

Further research was done on a range of soft algebraic structures in [29 ] ² [ 38] . Rao [39 ] ² [ 42] has developed several new semigroup types, including bi - interior ideals, bi - quasi - interior ideals, bi - quasi ideals, quasi - interior ideals, and weak interior ideals, extensions of existing ideals . Furthermore , Baupradist et al. [43] proposed the idea of essential ideals in semigroups .

Rao introduced the bi - quasi ideal of semigroups [41] as a generalization of bi - ideal and quasi - ideal. In contrast, the soft intersection bi - quasi ideal of semigroups was proposed in this paper to generalize the soft intersection bi - ideal and soft intersection quasi - ideal. In [10] , almost bi - quasi ideals are introduced as a further generalization of bi - quasi ideals defined in [41] . This study proposes the concept of soft intersection almost bi - quasi ideals and its generalization weakly - . Moreover, in contrast to soft intersection semigroup theory, our results show that every soft intersection almost bi - quasi ideal is also a soft intersection almost ideal and a soft intersection almost bi - ideal.

Furthermore, we show that an idempotent soft intersection almost bi - quasi ideal is a soft intersection almost subsemigroup, a soft intersection almost weak interior ideal, a soft intersection almost tri - ideal, and a soft intersection almost tri - bi - ideal. W e note that a semigroup may be constructed by soft intersection - s of a semigroup under the binary operation of soft union but not under the soft intersection operation. Also, by deriving that if a nonempty set is almost bi - l, then its soft characteristic function is also a soft intersection almost bi - quasi ideal, and vice versa, we establish the relationship between a semigroup's soft intersection almost bi - quasi ideal and almost bi - quasi ideal as regards minimality, primene ss, semiprimeness, and strongly primeness.

This section reviews several fundamental notions related to semigroups and soft sets. Definition 1. Let be the universal set, be the parameter set, ? ? be the power set of , and ? . A soft set ˆ ? over is a set - valued function such that ˆ ? ? ? ? ? such that for all š ? , ˆ ? ? š ? ? ? . A soft set over can be represented by the set of ordered pairs [1 3 ] , [2 4 ] . ˆ ?

? ? ? š ? ˆ ? ? š ? ? ? š ? ? ˆ ? ? š ? ? ? ? ? ? Throughout this paper, the set of all the soft sets over is designated by ? ? ? . Definition 2. Let ˆ ? ? ? ? ? . If ˆ ? ? š ? ? ? for all š ? , then ˆ ? is called a null soft set and denoted by ? ? . If ˆ ? ? š ? ? for all š ? , then ˆ ? is called an absolute soft set and is denoted by ? [ 24 ] . - Definition 3. Let ˆ ? ?

ˆ ? ? ? ? ? . If ˆ ? ? š ? ? ˆ ? ? š ? for all š ? , then ˆ ? is a soft subset of ˆ ? and denoted by ˆ ? ? ? ˆ ? . If ˆ ? ? š ? ? ˆ ? ? š ? for all š ? , then ˆ ? is called soft equal to ˆ ? and denoted by ˆ ? ? ˆ ? [2 4 ] . Definition 4. ‡– ˆ ? , ˆ ? ? ? ? ? . The union of ˆ ? and ˆ ? is the soft set ˆ ? ? ? ˆ ? , where ? ˆ ? ? ? ˆ ? ? ? š ? ? ˆ ? ? š ? ? ˆ ? ? š ? ? for all š ? . The intersection of ˆ ?

and ˆ ? is the soft set ˆ ? ? ? ˆ ? , where ? ˆ ? ? ? ˆ ? ? ? š ? ? ˆ ? ? š ? ? ˆ ? ? š ? ? for all š ? [2 4 ] . Definition 5. For a soft set ˆ ? , the support of ˆ ? is define d by [4 4 ] : •—’’ ? ˆ ? ? ? ? š ? ? ˆ ? ? š ? ? ? ? ? It is obvious that a soft set with an empty support is a null soft set; otherwise, the soft set is non - null. Note 1. If ˆ ? ? ? ˆ ? , then •—’’ ? ˆ ? ? ? •—’’ ?

ˆ ? ? [4 5 ] . A semigroup is a nonempty set with an associative binary operation, and throughout this paper, stands for a semigroup, and all the soft sets are the elements of ? ? ? unless otherwise specified. Definition 6 . A nonempty subset of is cal led : I. A left ( Right ) bi - quasi ideal of if ? ? ? ? ? ? , and a bi - quasi ideal of ? if is both a left bi - quasi ideal of and right bi - quasi ideal of [41] .

II. An almost left ( Right ) ideal of if ™ ? ? ? ? ™ ? ? ? ? ? for all ™ ? , and an almost ideal of if is both an almost left ideal of and an almost right ideal of [3] . III. An almost bi - ideal of if ™ ? ? ? ? for all ™ ? [4] . IV. An almost left ( Right ) bi - quasi ideal of if ? ™ ? š ? ? ? ? ? ? ™ ? š ? ? ? ? ? ? for all ™ ? š ? ? and an - Briefly almost BQ - ideal) of if is both an almost left bi - quasi ideal of and an almost right bi - quasi ideal of [10] .

V. A weakly almost left ( Right ) bi - quasi ideal of if ? ™ ? ™ ? ? ? ? ? ? ™ ? ™ ? ? ? ? ? ? for all ™ ? ? and a weakly almost bi - (Briefly weakly almost BQ - ideal) of if is both a weakly almost left bi - quasi ideal of and a weakly almost right bi - quasi ideal of [10] . Definition 7 . An almost left ( Right ) bi - of is called a minimal almost left ( Right ) bi - of if for any almost bi - of if whenever ?

, then ? [10] . Definition 8 . Let be an almost bi - quasi ideal of . Then, is called : I. A prime almost bi - if for any almost bi - and of such that ? implies that ? or ? . II. A semiprime almost bi - quasi ideal if for any almost bi - of such that ? implies that ? . III. A strongly prime almost bi - quasi ideal if for any almost bi - and of such that ?

? implies that ? or ? [10] . Definition 9 . Let ˆ ? and ‰ ? be soft sets over the common universe . Then, soft intersection product ˆ ? ? ‰ ? is defined by [2 6 ] . ? ? ˆ ? ? › ? ? ‰ ? ? œ ? ? ? ‹ˆ ? › ? œ ? •—…Š –Šƒ– š ? ›œ ? ? ˆ ? ? ‰ ? ? ? š ? ? ? ? ? ?? ? ? ‘–Š‡”™‹•‡ ? Theorem 1.

Le t „ ? , † ? ? Ž ? ? ? ? ? . Then, I. ? „ ? ? † ? ? ? Ž ? ? „ ? ? ? † ? ? Ž ? ? . II. „ ? ? † ? ? † ? ? „ ? ? ‰‡‡”ƒŽŽ› ? III. „ ? ? ? † ? ? ? Ž ? ? ? ? „ ? ? † ? ? ? ? ? „ ? ? Ž ? ? and ? „ ? ? ? † ? ? ? Ž ? ? ? „ ? ? Ž ? ? ? ? ? † ? ? Ž ? ? . Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3 IV. „ ? ? ? † ? ?

? Ž ? ? ? ? „ ? ? † ? ? ? ? ? „ ? ? Ž ? ? and ? „ ? ? ? † ? ? ? Ž ? ? ? „ ? ? Ž ? ? ? ? ? † ? ? Ž ? ? . V. If „ ? ? ? ? † ? ? then „ ? ? Ž ? ? † ? ? Ž ? and Ž ? ? „ ? ? ? Ž ? ? † ? . VI. If – ? ,  ? ? ? ? ? ? ? such that – ? ? „ ? and  ? ? † ? ? then – ? ?  ? ? ? „ ? ? † ? [2 6 ] . Definition 1 0 . Let be a subset of . We denote by ? the soft characteristic function of and defi ne as : ?

‹ˆ š ? ? ? ? š ? ? ? ? ? ‹ˆ š ? ? ? The soft characteristic function of is a soft set over , that is, ? : ? ? ? [2 6 ] . If ˆ ? ? š ? ? for all š ? , then we denote such a kind of soft set by ? ? throughout this paper. It is obvious that ? ? ? , that is, ? ? ? ? š ? ? for all š ? [2 6 ] . Corollary 1.

—’’ ? ? ? ? [4 5 ] . Theorem 2. Let and be nonempty subsets of . Then, the following properties hold [2 6 ] , [4 5 ] : I. ? if and only if ? ? ? ? . II. ? ? ? ? ? ? ? ? and ? ? ? ? ? ? ? ? . III. ? ? ? ? ?? . Definition 1 1 . Let š be an element in . We denote by ? the soft characteristic function of š and de fined as : ? ‹ˆ › ? š ?

? ? › ? ? ? ? ? ‹ˆ › ? š ? The soft characteristic function of š is a soft set over , that is, ? : ? ? ? [4 6 ] . Definition 1 2 . A soft set † ? of over is c alled : I. A soft intersection left ( Right ) bi - quasi ideal of over if ? ? ? ? † ? ? ? ? † ? ? ? ? † ? ? ? ? ? ? † ? ? ? † ? ? ? ? ? ? ? ? † ? ? ? ? ? † ? ? ? ? † ? ? , and a soft intersection bi - quasi ideal of if † ?

is both a soft intersection left bi - quasi ideal of and a soft intersection right bi - of . II. A soft intersection almost subsemigroup of over if ? † ? ? † ? ? ? ? † ? ? ? ? [4 5 ]. III. A soft intersection almost left ( Right) ideal of over if ? ? ? † ? ? ? ? † ? ? ? ? ? ? † ? ? ? ? ? ? † ? ? ? ? ? for all š ? , and a soft intersection almost ideal of if † ? is both a soft intersection almost left ideal of and a soft intersection almost right ideal of [4 6 ] .

IV. A - ideal of over if ? † ? ? ? ? † ? ? ? ? † ? ? ? ? for all š ? [4 7 ] . V. A soft intersection almost left (Right) weak interior ideal of over if ? ? ? † ? ? † ? ? ? ? † ? ? ? ? ? ? † ? ? † ? ? ? ? ? ? † ? ? ? ? ? for all š ? ? and a soft intersection almost weak interior ideal of if † ? is both a soft intersection almost left weak interior ideal of and a soft intersection almost right weak interior ideal of [4 8 ] .

VI. A soft intersection almost left ( Right ) tri - ideal of over if ? † ? ? ? ? † ? ? † ? ? ? ? † ? ? ? ? ? ? † ? ? † ? ? ? ? † ? ? ? ? † ? ? ? ? ? ? for all š ? , and a soft intersection almost tri - ideal of if † ? is both a soft intersection almost left tri - ideal of and a soft intersection almost right tri - ideal of [ 49 ] . VII. A soft intersection almost tri - bi - ideal of over if ?

† ? ? † ? ? ? ? † ? ? † ? ? ? ? † ? ? ? ? for all š ? [5 0 ] . It is easy to see that if † ? ? š ? ? for all š ? , then † ? is a soft intersection bi - quasi ideal of . As is mentioned above, we denote such a kind of soft intersection bi - by ? ? . Regarding the probable consequences of network analysis and graph applications concerning soft sets ( Defined by the divisibility of determinants), we refer to [5 1 ] .

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Definition 1 3 . A soft set ˆ ? is called a soft intersection almost left ( Right ) bi - quasi ideal of if ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? for a ll š ? › ? . ˆ ? is called a soft intersection, the almost bi - quasi ideal of if ˆ ? is both a soft intersection almost left bi - quasi ideal of and a soft intersection almost right bi - quasi ideal of .

A soft set ˆ ? is called a soft intersection weakly almost left (Right ) bi - quasi ideal of if ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? for all š ? . ˆ ? is called a soft intersection weakly almost bi - quasi ideal of if ˆ ? is both a weakly almost left bi - quasi ideal of and a soft intersection weakly almost right bi - quasi ideal of .

Hereafter, f or brevity, a soft intersection is designated by SI, and (Left /right) bi - quasi ideal is designed by (Left /right) BQ - ideal. Thus, soft intersection ( Weakly ) almost ( Left /right) bi - quasi ideal is denoted by SI - (Weakly ) almost ( Left /right) BQ - ideal. Here also note that since the operation of soft intersection is commutative in ? ? ? , it is obvious that in Definition 13 , ?

? ? ˆ ? ? and ? ˆ ? ? ? ? ˆ ? ? ? ‹‹Žƒ”Ž› ? ? ˆ ? ? ? ? ƒ† ? ˆ ? ? ? ? ˆ ? ? ? can commute with each other for all š ? › ? . Example 1. Consider the semigroup ? ? ? ? ? ? under the binary operation with the following table : Table 1. Cayley table of binary operation. ? ? ? ? ? ? ? ? ? ? Let ˆ ? , Š ? , and ‰ ? be soft sets over ? ? ? ? ? š ?

› ? ? š ? › ? ‡ ? š› ? ›š ? ? ? ‡ ? š ? › ? ›š ? as follows: ˆ ? ? ? ? ? ? ? ‡ ? š ? ? ? ? ? ? ? ‡ ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? › ? ›š ? ? ? ? ‰ ? ? ? ? ? ? ? š ? › ? ? ? ? ? ? ? ‡ ? ›š ? ? ? ? Here, ˆ ? and Š ? are both SI - ( Weakly ) almost BQ - ideal. Let's first show that ˆ ? is an SI - (Weakly ) almost left BQ - ideal, that is, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ?

? ? ˆ ? ? ? ? , for all š ? › ? . Let's start with ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? : ? ? ? ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ?

? ˆ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3 ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? Consequ ently, ?

? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ? ‡ ? š ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? Sim ilarly, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ?

? ? ? ? ? ‡ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? Therefore, ˆ ? is an SI - ( Weakly ) almost left BQ - ideal. And also ˆ ? is an SI - ( Weakly ) almost right BQ - ideal, that is, ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . In fac t; ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ? ‡ ? š ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ? ? ˆ ? ?

? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? ‡ ? ? ? ? ? ? ? Thus, ˆ ? is an SI - ( Weakly ) almost right BQ - ideal. Hence, ˆ ? is an SI - ( Weakl y) almost BQ - ideal.

Similarly, we can show that Š ? is an SI - ( Weakl y) almost left BQ - ideal and SI - ( Weak ly) almost right BQ - ideal. Let's first show that Š ? is an SI - ( Weak ly) almost left B Q - ideal: ? ? ? Š ? ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? › ?

›š ? ? ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? Consequently, Š ? is an SI - ( Weakl y) almost left BQ - ideal. Let's continue with Š ? is an SI - ( Weak ly) almost right B Q - ideal: ? ? Š ? ? ? ? ? ? ? Š ?

? ? ? ? Š ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? › ? ›š ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? ›š ? ? ? ? ? ? ? ›š ?

? ? ? ? ? ? Therefore, Š ? is an SI - ( Weakl y) almost right BQ - ideal. Thus, Š ? is an SI - ( Weakl y) almost BQ - ideal. One can also show that ‰ ? is not an SI - ( Weakl y) almost BQ - ideal. In fact; - ? ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ?

? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ?

‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? Consequ ently, ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Hence, ‰ ? is not an SI - ( Weakly) almost left BQ - ideal. Similarly since ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ‰ ? is not an SI - ( Weakly ) almost right BQ - ideal and ? ? is not an SI - ( Weakl y) almost BQ - ideal.

From now on, the proofs are given for only SI - almost left BQ - ideal, since the proofs for SI - almost ( Righ t) BQ - ideal can be shown similarly. Proposition 1. Every SI - almost BQ - ideal is an SI - weakly almost BQ - ideal. Proof: Let ˆ ? b e an SI - almost BQ - ideal. Then, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? and ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ?

ˆ ? ? ? ? ? ˆ ? ? ? ? for all š ? › ? . Hence, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? and ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? for all š ? . So, ˆ ? is an SI - weakly almost BQ - ideal. Since SI - weakly almost BQ - ideal is a generalization of SI - almost BQ - ideal, from now on, all the theorems and proofs are given for SI - almost BQ - ideal instead of SI - weakly almost BQ - ideals.

Proposition 2. Let ˆ ? be an SI - left ( Resp . right) BQ - ideal. ˆ ? is either ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? , for some š ? › ? or an SI - almost left ( Resp . right) BQ - ideal. Proof: Let ˆ ? be an SI - left BQ - ideal, then, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? and let ? ? ? ˆ ?

? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? . We need to show that ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? for all š ? › ? . Since ? ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? , it follows that ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ˆ ? . From assumption ? ? ? ˆ ? ? ? ? ? ˆ ? ?

? ? ˆ ? ? ? ? ? is obvious. Then, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? implyin g that ˆ ? is an SI - almost left BQ - ideal. Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3 Here it is obvious that, if ˆ ? is an SI - left BQ - ideal, and for some š ? › ? , ? ? ? ˆ ? ? ?

? ? ˆ ? ? ? ? ˆ ? ? ? ? ? then, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? . Therefore, ˆ ? is not an SI - almost left BQ - ideal. Corollary 2. If ˆ ? is an SI - almost left ( Resp . right) BQ - ideal, then ˆ ? needs not be an SI - left ( Resp . right) BQ - ideal. Example 2. In Example 1 , it is shown that Š ? is an SI - almost left BQ - ideal; however Š ?

is not an SI - left bi - . In fact, ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? Š ? ? ? ? ? Hence, Š ? is not an SI - left BQ - ideal. Similarly, Š ? is an SI - almost right BQ - ideal; however, Š ? is not an SI - right BQ - ideal. In fact, ?

? Š ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? ? Š ? ? ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? ? ? Š ? ? ? ? ? Hence, Š ? is not an SI - right BQ - ideal. Theorem 3 . Every SI - almost left ( Resp . right) BQ - ideal is an SI - almost left ( Resp . right) ideal. Proof: Assume that ˆ ? is an SI - almost left BQ - ideal. H ence, ?

? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? f or al l š ? › ? . We need to show that ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? , for all š ? . ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? Since ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Hence, ˆ ?

is an SI - almost left ideal. Theorem 4. Every SI - almost (Left /right) BQ - ideal is an SI - - ideal. Proof: Assume that ˆ ? is an SI - almost left BQ - ideal. Hence, ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . We need to show that ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ?

? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? Since ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Hence, ˆ ? is an SI - - ideal. The following example shows that the converse of Theorem 4 is not true in general: Example 3. Consider the soft set ‰ ? in Example 1 . Here, ‰ ? is an SI - - ideal, that is, ?

‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? , for all š ? . Let's first show that ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? : - ? ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ?

? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ? ‰ ? ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ? ‰ ? ? ? ? ? ‰ ? ? ? ? ? Consequently, ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ? š ? › ? ? ? ? ? ? ? ? ? ? ? ? ? Similarly, ? ‰ ? ? ? ? ‰ ? ? ? ? ‰ ? ? ? ? ? ? ?

? ? ? ? ? ? ‡ ? ›š ? ? ? ? ? ? ? Thus, ‰ ? is an SI - - ideal. However, it is clear that ‰ ? is not an SI - almost (Left /right) BQ - ideal, as seen in Example 1 . Proposition 3. Let ˆ ? be an idempotent soft set. If ˆ ? is an SI - almost ( Left /right) BQ - ideal, then ˆ ? is an SI - almost subsemigroup. Proof: Assume that ˆ ? is an SI - almost left BQ - ideal such that ˆ ? is an idempotent, then ˆ ?

? ˆ ? ? ˆ ? and ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . We need to show that ˆ ? is an SI - almost subsemigroup, that is ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? . ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ˆ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ˆ ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ?

? ? ˆ ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ˆ ? ? ? ˆ ? ? Since ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Thus, ˆ ? is an SI - almost subsemigroup. Proposition 4. Let ˆ ? be an idempotent soft set. If ˆ ? is an SI - almost left ( Resp . right) BQ - ideal, then ˆ ? is an SI - almost left ( Resp . right) weak interior ideal.

Proof: Assume that ˆ ? is an SI - almost left BQ - ideal such that ˆ ? is an idempotent, then ˆ ? ? ˆ ? ? ˆ ? and ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . We need to show that ˆ ? is an SI - almost left weak interior ideal, that is ? ? ˆ ? ? ˆ ? ? ? ˆ ? ? ? ? , for all š ? . ? ? ? ? ˆ ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ˆ ?

? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ˆ ? ? ? ? ? ˆ ? ? ˆ ? ? ? ˆ ? ? Since ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Thus, ˆ ? is an SI - almost left weak interior ideal. Proposition 5. Let ˆ ? be an idempotent soft set. If ˆ ? is an SI - almost left (Right ) BQ - ideal, then ˆ ? is an SI - almost tri - ideal.

Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3 Proof: Assume that ˆ ? is an idempotent SI - almost left BQ - ideal such that ˆ ? ? ˆ ? ? ˆ ? and ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? for all š ? › ? . We need to show that ˆ ? is an SI - almost tri - ideal, that is ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? and ? ˆ ? ?

ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? , for all š ? . ? ? ˆ ? ? ˆ ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ˆ ? ? ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ˆ ? ? ? Since ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Hence, ˆ ? is an SI - almost left tri - ideal.

? ? ˆ ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ˆ ? ? ? ? ˆ ? ? ? ˆ ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ˆ ? ? Since ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Hence, ˆ ? is an SI - almost right tri - ideal. Thus, ˆ ? is an SI - almost tri - ideal. Proposition 6.

Let ˆ ? be an idempotent soft set. If ˆ ? is an SI - almost (Left /right) BQ - ideal, then ˆ ? is an SI - almost tri - bi - ideal. Proof: Assume that ˆ ? is an SI - almost left BQ - ideal such that ˆ ? is an idempotent, then ˆ ? ? ˆ ? ? ˆ ? and ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? for all š ? › ? . We need to show that ˆ ? is an SI - almost tri - bi - ideal, that is ?

ˆ ? ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? for all š ? . ? ? ˆ ? ? ˆ ? ? ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? Since ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ˆ ? ? ? ? ˆ ? ? ? ? . Thus, ˆ ?

is an SI - almost tri - bi - ideal. Theorem 5 . Let ? ? ? ? ? ? . If ˆ ? is an SI - almost left ( Resp . right) BQ - ideal, then Š ? is an SI - almost left ( Resp . right) BQ - ideal. Proof: Assume that ˆ ? is an SI - almost left BQ - ideal. Hence, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . We need to show that ? ? ? ? Š ? ? ? ? ? Š ? ? ?

? Š ? ? ? ? ? Š ? ? ? ? , for all š ? › ? , In fact, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? Since ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , it is obvious that ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? Š ? ? ? ? ? Š ? ? ? ? . This completes the proof. Theorem 6.

If ˆ ? and Š ? are SI - almost left ( Resp . right) BQ - ideals, then ˆ ? ? ? Š ? is an SI - almost left ( Resp . right) BQ - ideal. Proof: Since ˆ is an SI - almost left BQ - ideal and ˆ ? ? ? ? ˆ ? ? ? Š ? , ˆ ? ? ? Š ? is an SI - almost left BQ - ideal by Theorem 5 . Corollary 3. The finite union of SI - almost left ( Resp . right) BQ - ideals is an SI - almost left ( Resp . right) BQ - ideal.

Corollary 4. Let ˆ ? or Š ? be an SI - almost left ( Resp . right) BQ - ideal, then ˆ ? ? ? Š ? is an SI - almost left ( Resp . right) BQ - ideal. Here, note that if ˆ ? and Š ? are SI - almost left ( Resp . right) BQ - ideals, then ˆ ? ? ? Š ? needs not be an SI - almost left ( Resp . right) BQ - ideal. Example 4. Consider the SI - almost left ( Resp . right) BQ - ideal ˆ ? and Š ? in Example 1 . Since, ˆ ?

? ? Š ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ˆ ? ? ? Š ? is not an SI - almost left ( Resp . right) BQ - ideal. - Now, we give the relationship between almost BQ - ideal and SI - almost BQ - ideal. But first of all, we remind the following lemma in order to use it in Theorem 7 . Lemma 1. Let š ? and be a nonempty subset of . Then, ? ? ? ? ?? . If is a nonempty subset of and › ?

, then ? ? ? ? ?? [4 6 ] . Theorem 7 . Let be a nonempty subset of . Then, is an almost left ( Resp . right) BQ - ideal if and only if ? , the soft characteristic function of , is an SI - almost left ( Resp . right) BQ - ideal. Proof: Assume that ? ? is an almost left BQ - ideal. Then, ? š ? › ? ? ? ? , for all š ? › ? , and so there exist Œ ? such that Œ ? ? š ?

œ ? ? . Since, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Œ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ??? ? ? ? Œ ? ? ? ? ? ?? ??? ? ? ? ? ? ? ? Œ ? ? ? ? ?? ? ??? ? ? ? ? Œ ? ? ? ? ? ? ? It follows that ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . Thus, ? is an SI - almost left BQ - ideal. Conversely, assume that ? is an SI - almost left BQ - ideal. Hence, we have ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? , for all š ? › ? . In order to show that is an almost left BQ - ideal, we should prove that ? ? and ? š ? › ? ? ? ? , for all š ? › ? . ? ? is obvious from the assumption. Now, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??? ? ? ? ? Hence, ? š ? › ? ? ? ? . Consequently, is an almost left BQ - ideal. Lemma 2. Let ˆ ? ? ? ? ? . Then, ˆ ? ? ? ???? ? ? ? ? [4 5 ] . Theorem 8. If ˆ ? is an SI - almost left ( Resp . right) BQ - ideal, then •—’’ ?

ˆ ? ? is an almost left ( Resp . right) BQ - ideal. Proof: Assume that ˆ ? is an SI - almost left BQ - ideal. Thus, ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? , for all š ? › ? . In order to show that •—’’ ? ˆ ? ? is an almost left BQ - ideal, by Theorem 8 , it is enough to show that ???? ? ? ? ? is an SI - almost left BQ - ideal. By Lemma 2 , ? ? ? ? ˆ ? ? ? ? ?

ˆ ? ? ? ˆ ? ? ? ? ? ? ˆ ? ? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ???? ? ? ? ? ? ? ???? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? and ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ˆ ? ? ? ? ? ˆ ? ? ? ? ? it implies that ? ? ? ? ???? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ? ? ???? ? ? ? ? . Consequently, is an SI - almost left BQ - ? ? ? ????

? ? ? ? ideal and by Theorem 7 , •—’’ ? ˆ ? ? is an almost left BQ - ideal. The following example shows that the converse of Theorem 8 is not true in general: Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3 Example 5. We know that ‰ ? is not an SI - almost left BQ - ideal in Example 1 . Since •—’’ ? ‰ ? ? ? ? ? ? ? ? , ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? •—’’ ? ‰ ? ? is an almost left BQ - ideal. Similarly, ‰ ? is not an SI - almost left BQ - ideal, but since ? ? ? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? •—’’ ? ‰ ? ? is an almost right BQ - ideal. That is to say, •—’’ ? ‰ ? ? is an almost BQ - ideal, although ‰ ?

is not an SI - almost BQ - ideal. Definition 1 4 . Let ˆ ? and Š ? be SI - almost left ( Resp . right) BQ - ideals such that Š ? ? ? ˆ ? . If •—’’ ? Š ? ? ? •—’’ ? ˆ ? ? , then ˆ ? is called a minimal SI - almost BQ - ideal. Theorem 9. Let be a nonempty subset of . Then, is a minimal almost left ( Resp . right) BQ - ideal if and only if ? , the soft characteristic function of , is a minimal SI - almost left ( Resp . right) BQ - ideal.

Proof: Assume that is a minimal, almost left BQ - ideal. Thus, is an almost left BQ - ideal, and so ? is an SI - almost left BQ - ideal by Theorem 7 . Let ˆ ? be an SI - almost left BQ - ideal such that ˆ ? ? ? ? . By Theorem 8 , •—’’ ? ˆ ? ? is an almost left BQ - ideal and by Note 1 and Corollary 1 , •—’’ ? ˆ ? ? ? •—’’ ? ? ? ? ? Since is a minimal almost left BQ - ideal, •—’’ ?

ˆ ? ? ? •—’’ ? ? ? ? . Thus, ? is a minimal SI - almost left BQ - ideal by Definition 14 . Conversely, let ? be a minimal SI - almost left BQ - ideal. Thus, ? is an SI - almost left BQ - ideal, and is an almost left BQ - ideal by Theorem 7 . Let be an almost left BQ - ideal such that ? . By Theorem 7 , ? is an SI - almost left BQ - ideal, and by Theorem 2 (i) , ? ? ? ? . Since ?

is a minimal SI - almost left BQ - ideal, ? •—’’ ? ? ? ? •—’’ ? ? ? ? ? by Corollary 1 . Thus , is a minimal, almost left BQ - ideal. Definition 15. Let ˆ ? , ‰ ? ? and Š ? be any SI - almost left ( Resp . right) BQ - ideals. If Š ? ? ‰ ? ? ? ˆ ? implies that Š ? ? ? ? ˆ ? or ‰ ? ? ˆ ? , then ˆ ? is called an SI - prime almost left ( Resp . right) BQ - ideal. Definition 16.

Let ˆ ? and Š ? be any SI - almost left ( Resp . right) BQ - ideals. If Š ? ? Š ? ? ? ˆ ? implies that Š ? ? ? ˆ ? , then ˆ ? is called an SI - semiprime almost left ( Resp . right) BQ - ideal. Definition 17. Let ˆ ? ? ‰ ? ? and Š ? be any SI - almost left ( Resp . right) BQ - ideals. If ? Š ? ? ‰ ? ? ? ? ? ‰ ? ? Š ? ? ? ? ˆ ? implies that Š ? ? ? ˆ ? or ‰ ? ? ? ˆ ? , then ˆ ?

is called an SI - strongly prime almost left ( Resp . right) BQ - ideal. It is obvious that every SI - strongly prime almost left ( Resp . right) BQ - ideal is an SI - prime almost left ( Resp . right) BQ - ideal, and every SI - prime almost left ( Resp . right) BQ - ideal is an SI - semiprime almost left ( Resp . right) BQ - ideal. Theorem 10. If ? , the soft characteristic function of , is an SI - prime almost left ( Resp . right) BQ - ideal, then is a prime almost left ( Resp . right) BQ - ideal, where ?

? ? . Proof: Assume that ? is an SI - prime almost left BQ - ideal. Thus, ? is an SI - almost left BQ - ideal, and thus, is an almost left BQ - ideal by Theorem 7 . Let and be almost left BQ - ideals such that ? . Thus, by Theorem 7 , ? and ? are SI - almost left BQ - ideals, and by Theorem 2 (i) and (iii) , - ? ? ? ? ? ?? ? ? ? Since ? is an SI - prime almost left BQ - ideal and ?

? ? ? ? ? , it follows that ? ? ? or ? ? ? ? ? . Therefore, by Theorem 2 (i) , ? or ? . Consequently, is a prime almost left BQ - ideal. Theorem 11. If ? , the soft characteristic function of , is an SI - semiprime almost left ( Resp . right) BQ - ideal, then is a semiprime almost left ( Resp . right) BQ - ideal, where ? ? ? . Proof: Assume that ? is an SI - semiprime almost left BQ - ideal. Thus, ?

is an SI - almost left BQ - ideal, and thus, is an almost left BQ - ideal by Theorem 7 . Let be an almost left BQ - ideal such that ? . Thus, by Theorem 7 , ? is an SI - almost left BQ - ideal, and by Theorem 2 (i) and (ii) , ? ? ? ? ? ?? ? ? Since ? is an SI - prime almost left BQ - ideal and ? ? ? ? ? ? ? , it follows that ? ? ? . Therefore , by Theorem 2 (i)

? . Consequently, is a semiprime almost left BQ - ideal. Theorem 12. If ? , the soft characteristic function of , is an SI - strongly prime almost left ( Resp . right) BQ - ideal, then is a strongly prime almost left ( Resp . right) BQ - ideal, where ? ? ? ? ? . Proof: Assume that ? is an SI - strongly prime almost left BQ - ideal. Thus, ? is an SI - almost left BQ - ideal, and thus, is an almost left BQ - ideal by Theorem 7 . Let and be almost left BQ - ideals such that ?

? . Thus, by Theorem 7 , ? and ? are SI - almost left BQ - ideals, and by Theorem 2 , ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ? ?? ? ?? ? ? ? Since ? ? ? is an SI - strongly prime almost left BQ - ideal and ? ? ? ? ? ? ? ? ? ? ? ? ? ? , it follows that ? ? ? or ? ? ? ? . Thus, by Theorem 2 (i) , ? or ? . Therefore, is a strongly prime, almost left BQ - ideal.

The concepts of "soft intersection almost bi - quasi ideal" and "soft intersection weakly almost bi - of semigroups were defined in this work. We demonstrated that although any soft intersection almost bi - quasi ideal is also a soft intersection weakly almost bi - quasi ideal, a soft intersection almost ideal, and a soft - ideal of a semigroup; the converses are not true in general with counterexamples.

Additionally, it was shown that an idempotent soft intersection almost bi - quas i ideal is a soft intersection almost subsemigroup, a soft intersection almost weak interior ideal, a soft intersection almost tri - ideal, and a soft intersection almost tri - bi - ideal. We obtained the relation between soft intersection almost bi - of a semigroup and almost bi - quasi ideal of a semigroup according to minimality, primeness, semiprimeness, and strongly primeness with the obtained theorem that if a nonempty set is almost bi - quasi ideal then its soft characteristic function is soft inte - , and vice versa. Additionally, we investigated that, unlike soft intersection operation, soft union operation can form a semigroup with the collection of almost bi - of a semigroup.

In future studies, some kinds of semigroup ideals, such as quasi - interior ideals, bi - interior ideals, and bi - quasi - interior ideals, may be studied in terms of soft intersection almost ideals.

This paper is derived from the second author's Master Thesis, supervised by the first author at Amasya University, Türkiye.

The authors conducted this research independently without external financial support. All costs associated with this study were covered by the researchers' institutional resources. Sezgin and ùOJLQ | Soft. Comput. Fusion. Appl. 1 ( 1 ) ( 2024 ) 2 8 - 4 3

The authors affirm that there are no competing interests ³ financial, professional, or personal ³ that could influence the objectivity or outcomes of this research. This study was conducted with full academic ntegrity, free from any affiliations or relationshi ps that might pose a conflict.

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