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 models 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 probability 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.

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 2021, Chinram and Nakkhasen [8] in 2022, Gaketem [9] in 2022, and Gaketem and Chinram [10] in 2023, Respectively, subsequently introduced almost subsemigroups, almost bi-quasi-interior ideals, almost bi-interior ideals, and almost bi-quasi ideals of semigroups. Additionally, several almost fuzzy semigroup ideal types were investigated in [5], [7]–[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]–[23], the fundamental operations of soft sets were examined. Çağman and Enginoğlu [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]–[38]. Rao [39]–[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 soft intersection weakly almost bi-quasi ideals of semigroups. 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. We note that a semigroup may be constructed by soft intersection almost bi-quasi ideals 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 A is almost bi-quasi ideal, 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, primeness, semiprimeness, and strongly primeness.

This section reviews several fundamental notions related to semigroups and soft sets.

Definition 1. Let U be the universal set, E be the parameter set, P(U) be the power set of U, and K ⊆ E. A soft set fK over U is a set-valued function such that fK: E → P(U) such that for all x ∉ K, fK(x) = ∅. A soft set over U can be represented by the set of ordered pairs [13], [24].

fK = {(x, fK(x)): x ∈ E, fK(x) ∈ P(U)}.

Throughout this paper, the set of all the soft sets over U is designated by SE(U).

Definition 2. Let fA ∈ SE(U). If fA(x) = ∅ for all x ∈ E, then fA is called a null soft set and denoted by ∅E. If fA(x) = U for all x ∈ E, then fA is called an absolute soft set and is denoted by UE [24].

Definition 3. Let fA, fB ∈ SE(U). If fA(x) ⊆ fB(x) for all x ∈ E, then fA is a soft subset of fB and denoted by f ⊆ ̃ f . If f (x) = f (x) for all x ∈ E, then f is called soft equal to f and denoted by f = f [24].

A B A B A B A B

Definition 4. Let f , f ∈ S (U). The union of f and f is the soft set f ∪ ̃ f , where (f ∪ ̃ f )(x) = f (x) ∪

A B E A B A B A B A

f (x), for all x ∈ E. The intersection of f and f is the soft set f ∩ ̃ f , where (f ∩ ̃ f )(x) = f (x) ∩ f (x), for

B A B A B A B A B

all x ∈ E [24].

Definition 5. For a soft set fA, the support of fA is defined by [44]:

supp(fA) = {x ∈ A: fA(x) ≠ ∅}.

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 f ⊆ ̃ f , then supp(f ) ⊆ supp(f ) [45].

A B A B

A semigroup S is a nonempty set with an associative binary operation, and throughout this paper, S stands for a semigroup, and all the soft sets are the elements of SS(U) unless otherwise specified.

Definition 6. A nonempty subset F of S is called:

I. A left (Right) bi-quasi ideal of S if SF ∩ FSF ⊆ F (FS ∩ FSF ⊆ F), and a bi-quasi ideal of if F is both a left bi-quasi ideal of S and right bi-quasi ideal of S [41].

II. An almost left (Right) ideal of S if wF ∩ F ≠ ∅ (Fw ∩ F ≠ ∅), for all w ∈ S, and an almost ideal of S if F is both an almost left ideal of S and an almost right ideal of S [3].

III. An almost bi-ideal of S if FwF ∩ F ≠ ∅, for all w ∈ S [4].

IV. An almost left (Right) bi-quasi ideal of S if (wF ∩ FxF) ∩ F ≠ ∅ ((Fw ∩ FxF) ∩ F ≠ ∅), for all w, x ∈ S, and an almost bi-quasi ideal (Briefly almost BQ-ideal) of S if F is both an almost left bi-quasi ideal of S and an almost right bi-quasi ideal of S [10].

V. A weakly almost left (Right) bi-quasi ideal of S if (wF ∩ FwF) ∩ F ≠ ∅ ((Fw ∩ FwF) ∩ F ≠ ∅), for all w ∈ S, and a weakly almost bi-quasi ideal (Briefly weakly almost BQ-ideal) of S if F is both a weakly almost left bi-quasi ideal of S and a weakly almost right bi-quasi ideal of S [10].

Definition 7. An almost left (Right) bi-quasi ideal A of S is called a minimal almost left (Right) bi-quasi ideal of S if for any almost bi-quasi ideal B of S if whenever B ⊆ A, then A = B [10].

Definition 8. Let P be an almost bi-quasi ideal of S. Then, P is called:

I. A prime almost bi-quasi ideal if for any almost bi-quasi ideals A and B of S such that AB ⊆ P implies that A ⊆ P or B ⊆ P.

II. A semiprime almost bi-quasi ideal if for any almost bi-quasi ideal A of S such that AA ⊆ P implies that A ⊆ P. III. A strongly prime almost bi-quasi ideal if for any almost bi-quasi ideals A and B of S such that AB ∩ BA ⊆ P implies that A ⊆ P or B ⊆ P [10].

Definition 9. Let fS and gS be soft sets over the common universe U. Then, soft intersection product fS ° gS is defined by [26].

⋃ {fS(y) ∩ gS(z)}, if ∃y, z ∈ S such that x = yz, (fS ° gS)(x) = {

x=yz

∅, otherwise.

Theorem 1. Let bS, dS, lS ∈ SS(U). Then, I. (bS ° dS) ° lS = bS ° (dS ° lS).

II. bS ° dS ≠ dS ° bS , generally.

III. b ° (d ∪ ̃ l ) = (b ° d ) ∪ ̃ (b ° l ) and (b ∪ ̃ d ) ° l = (b ° l ) ∪ ̃ (d ° l ).

S S S S S S S S S S S S S S IV. b ° (d ∩ ̃ l ) = (b ° d ) ∩ ̃ (b ° l ) and (b ∩ ̃ d ) ° l = (b ° l ) ∩ ̃ (d ° l ).

S S S S S S S S S S S S S S

V. If b ⊆ ̃ d , then b ° l ⊆ ̃ d ° l and l ° b ⊆ ̃ l ° d .

S S S S S S S S S S

VI. If t , k ∈ S (U) such that t ⊆ ̃ b and k ⊆ ̃ d , then t ° k ⊆ ̃ b ° d [26].

S S S S S S S S S S S

Definition 10. Let A be a subset of S. We denote by SA the soft characteristic function of A and define as: U, if x ∈ A, SA(x) = { ∅, if x ∈ S\A.

The soft characteristic function of A is a soft set over U, that is, SA: S ⟶ P(U) [26].

If f (x) = U for all x ∈ S, then we denote such a kind of soft set by ̃ throughout this paper. It is obvious that ̃ = S , that is, ̃(x) = U for all x ∈ S [26].

Corollary 1. Supp(SA) = A [45].

Theorem 2. Let X and Y be nonempty subsets of S. Then, the following properties hold [26], [45]:

I. X ⊆ Y if and only if S ⊆ ̃ S .

II. S ∩ ̃ S = S and S ∪ ̃ S = S .

X Y X∩Y X Y X∪Y

III. SX ° SY = SXY.

Definition 11. Let x be an element in S. We denote by Sx the soft characteristic function of x and defined as: U, if y = x, Sx(y) = { ∅, if y ≠ x.

The soft characteristic function of x is a soft set over U, that is, Sx: S ⟶ P(U) [46].

Definition 12. A soft set dS of S over U is called:

I. A soft intersection left (Right) bi-quasi ideal of S over U if (̃ ° d ) ∩ ̃ (d ° ̃ ° d ) ⊆ ̃ d ((d ° ̃) ∩ ̃ (d ° ̃ ° d ) ⊆ ̃ d ), and a soft intersection bi-quasi ideal of S if d

S S S S S S S S S

is both a soft intersection left bi-quasi ideal of S and a soft intersection right bi-quasi ideal of S.

II. A soft intersection almost subsemigroup of S over U if (d ° d ) ∩ ̃ d ≠ ∅ [45].

S S S S

III. A soft intersection almost left (Right) ideal of S over U if (S ° d ) ∩ ̃ d ≠ ∅ ((d ° S ) ∩ ̃ d ≠ ∅ ) for all x ∈

x S S S S x S S

S, and a soft intersection almost ideal of S if dS is both a soft intersection almost left ideal of S and a soft intersection almost right ideal of S [46].

IV. A soft intersection almost bi-ideal of S over U if (d ° S ° d ) ∩ ̃ d ≠ ∅ for all x ∈ S [47].

S x S S S

V. A soft intersection almost left (Right) weak interior ideal of S over U if (S ° d ° d ) ∩ ̃ d ≠

x S S S

∅ ((d ° d ° S ) ∩ ̃ d ≠ ∅ ) for all x ∈ S, and a soft intersection almost weak interior ideal of S if d is both

S S S x S S S

a soft intersection almost left weak interior ideal of S and a soft intersection almost right weak interior ideal of S [48].

VI. A soft intersection almost left (Right) tri-ideal of S over U if (d ° S ° d ° d ) ∩ ̃ d ≠

S x S S S

∅ ((d ° d ° S ° d ) ∩ ̃ d ≠ ∅ ), for all x ∈ S, and a soft intersection almost tri-ideal of S if d is both a soft

S S S x S S S S

intersection almost left tri-ideal of S and a soft intersection almost right tri-ideal of S [49].

VII. A soft intersection almost tri-bi-ideal of S over U if (d ° d ° S ° d ° d ) ∩ ̃ d ≠ ∅ for all x ∈ S [50].

S S x S S S S

It is easy to see that if dS(x) = U for all x ∈ S, then dS is a soft intersection bi-quasi ideal of S. As is mentioned above, we denote such a kind of soft intersection bi-quasi ideal by ̃. Regarding the probable consequences of network analysis and graph applications concerning soft sets (Defined by the divisibility of determinants), we refer to [51].

Definition 13. A soft set fS is called a soft intersection almost left (Right) bi-quasi ideal of S if [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ ([(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ ),

x S S y S S S S x S y S S S

for all x, y ∈ S. fS is called a soft intersection, the almost bi-quasi ideal of S if fS is both a soft intersection almost left bi-quasi ideal of S and a soft intersection almost right bi-quasi ideal of S.

A soft set fS is called a soft intersection weakly almost left (Right) bi-quasi ideal of S if [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ ([(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ ),

x S S x S S S S x S x S S S

for all x ∈ S. fS is called a soft intersection weakly almost bi-quasi ideal of S if fS is both a soft intersection weakly almost left bi-quasi ideal of S and a soft intersection weakly almost right bi-quasi ideal of S.

Hereafter, for 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 SE(U), it is obvious that in Definition 13, (Sx ° fS) and (fS ° Sy ° fS) (Similarly, (fS ° Sx) and (fS ° Sy ° fS)) can commute with each other for all x, y ∈ S.

Example 1. Consider the semigroup S = { , } under the binary operation with the following table:

Let fS, hS, and gS be soft sets over U = D2 = {⟨x, y⟩: x = y = e, xy = yx} = {e, x, y, yx} as follows:

fS = {( , {e, x}), ( , {e})}.

hS = {( , {yx}), ( , {y, yx})}.

gS = {( , {x, y}), ( , {e, yx})}.

Here, fS and hS are both SI-(Weakly) almost BQ-ideal. Let's first show that fS is an SI-(Weakly) almost left BQ-ideal, that is, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S.

Let's start with [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f :

S S S S

[[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ]( ) = [(S ° f ) ∩ ̃ (f ° S ° f )]( ) ∩ f ( ),

S S S S S S S S

= (S ° fS)( ) ∩ (fS ° S ° fS)( ) ∩ fS( ), = fS( ) ∩ [fS( ) ∪ fS( )] ∩ fS( ), = fS( ),

[[( ° ) ∩ ̃ ( ° ° )] ∩ ̃ ]( ) = [( ° ) ∩ ̃ ( ° ° )]( ) ∩ ( )

= (S ° fS)( ) ∩ (fS ° S ° fS)( ) ∩ fS( ) = fS( ) ∩ [fS( ) ∩ fS( )] ∩ fS( ) = fS( ) ∩ fS( ).

Consequently, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e, x}), ( , {e})} ≠ ∅ .

S S S S S

Similarly, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e}), ( , {e})} ≠ ∅ ,

S S S S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e}), ( , {e})} ≠ ∅ ,

S S S S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e}), ( , {e})} ≠ ∅ .

S S S S S

Therefore, fS is an SI-(Weakly) almost left BQ-ideal. And also fS is an SI-(Weakly) almost right BQ-ideal, that is, [(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. In fact;

S x S y S S S

[(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e, x}), ( , {e})} ≠ ∅ ,

S S S S S

[(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e}), ( , {e})} ≠ ∅ ,

S S S S S

[( ° ) ∩ ̃ ( ° ° )] ∩ ̃ = {( , { }), ( , { })} ≠ ∅ , [(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = {( , {e}), ( , {e})} ≠ ∅ .

S S S S S

Thus, fS is an SI-(Weakly) almost right BQ-ideal. Hence, fS is an SI-(Weakly) almost BQ-ideal.

Similarly, we can show that hS is an SI-(Weakly) almost left BQ-ideal and SI-(Weakly) almost right BQ-ideal. Let's first show that hS is an SI-(Weakly) almost left BQ-ideal:

[(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {yx})} ≠ ∅ ,

S S S S S

[(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {y, yx})} ≠ ∅ ,

S S S S S

[(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {yx})} ≠ ∅ ,

S S S S S

[(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {yx})} ≠ ∅ .

S S S S S

Consequently, hS is an SI-(Weakly) almost left BQ-ideal. Let's continue with hS is an SI-(Weakly) almost right BQ-ideal:

[(h ° S ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {yx})} ≠ ∅ ,

S S S S S

[(h ° S ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {y, yx})} ≠ ∅ ,

S S S S S

[(h ° S ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {yx})} ≠ ∅ ,

S S S S S

[(h ° S ) ∩ ̃ (h ° S ° h )] ∩ ̃ h = {( , {yx}), ( , {yx})} ≠ ∅ .

S S S S S

Therefore, hS is an SI-(Weakly) almost right BQ-ideal. Thus, hS is an SI-(Weakly) almost BQ-ideal.

One can also show that gS is not an SI-(Weakly) almost BQ-ideal. In fact; [[(S ° g ) ∩ ̃ (g ° S ° g )] ∩ ̃ g ] ( ) = [(S ° g ) ∩ ̃ (g ° S ° g )]( ) ∩ g ( ),

S S S S S S S S

= (S ° gS)( ) ∩ (gS ° S ° gS)( ) ∩ gS( ), = gS( ) ∩ [gS( ) ∩ gS( )] ∩ gS( ),

= gS( ) ∩ gS( ),

= ∅ .

[[(S ° g ) ∩ ̃ (g ° S ° g )] ∩ ̃ g ] ( ) = [(S ° g ) ∩ ̃ (g ° S ° g )]( ) ∩ g ( ),

S S S S S S S S

= (S ° gS)( ) ∩ (gS ° S ° gS)( ) ∩ gS( ), = gS( ) ∩ [gS( ) ∪ gS( )] ∩ gS( ), = gS( ) ∩ gS( ), = ∅S.

Consequently, [(S ° g ) ∩ ̃ (g ° S ° g )] ∩ ̃ g = {( , ∅), ( , ∅)} = ∅ .

S S S S S

Hence, gS is not an SI-(Weakly) almost left BQ-ideal. Similarly since [(g ° S ) ∩ ̃ (g ° S ° g )] ∩ ̃ g {( , ∅), ( , ∅)} = ∅ .

S S S S= S

gS is not an SI-(Weakly) almost right BQ-ideal and is not an SI-(Weakly) almost BQ-ideal.

From now on, the proofs are given for only SI-almost left BQ-ideal, since the proofs for SI-almost (Right) BQ-ideal can be shown similarly.

Proposition 1. Every SI-almost BQ-ideal is an SI-weakly almost BQ-ideal.

Proof: Let fS be an SI-almost BQ-ideal. Then, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ and [(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ for all x, y ∈ S. Hence,

x S S y S S S S x S y S S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ and [(f ° S ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ for all x ∈ S.

x S S x S S S S x S x S S S

So, fS 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 f be an SI-left (Resp. right) BQ-ideal. f is either (S ° f ) ∩ ̃ (f ° S ° f ) =

S S x S S y S

∅ ((f ° S ) ∩ ̃ (f ° S ° f ) = ∅ ), for some x, y ∈ S or an SI-almost left (Resp. right) BQ-ideal.

S S x S y S S

Proof: Let f be an SI-left BQ-ideal, then, (̃ ° f ) ∩ ̃ (f ° ̃ ° f ) ⊆ ̃ f and let (S ° f ) ∩ ̃ (f ° S ° f ) ≠ ∅ . We

S S S S S x S S y S S

need to show that [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. Since (S ° f ) ∩ ̃ (f ° S ° f ) ⊆ ̃ (̃ ° f ) ∩ ̃ (f ° ̃ ° f ) ⊆ ̃ f , it follows that

x S S y S S S S S

(S ° f ) ∩ ̃ (f ° S ° f ) ⊆ ̃ f . From assumption (S ° f ) ∩ ̃ (f ° S ° f ) ≠ ∅ is obvious. Then,

x S S y S S x S S y S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = (S ° f ) ∩ ̃ (f ° S ° f ) ≠ ∅ .

x S S y S S x S S y S S

implying that fS is an SI-almost left BQ-ideal.

Here it is obvious that, if f is an SI-left BQ-ideal, and for some x, y ∈ S, (S ° f ) ∩ ̃ (f ° S ° f ) = ∅ then,

S x S S y S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = ∅ ∩ ̃ f = ∅ . Therefore, f is not an SI-almost left BQ-ideal.

x S S y S S S S S S

Corollary 2. If fS is an SI-almost left (Resp. right) BQ-ideal, then fS needs not be an SI-left (Resp. right) BQ-ideal.

Example 2. In Example 1, it is shown that hS is an SI-almost left BQ-ideal; however hS is not an SI-left bi-quasi ideal. In fact, [(̃ ° h ) ∩ ̃ (h ° ̃ ° h )]( ) = (̃ ° h )( ) ∩ (h ° ̃ ° h )( ),

S S S S S S

= [hS( ) ∪ hS( )] ∩ [hS( ) ∪ hS( )], = hS( ) ∪ hS( ), ⊈ hS( ).

Hence, hS is not an SI-left BQ-ideal. Similarly, hS is an SI-almost right BQ-ideal; however, hS is not an SI-right BQ-ideal. In fact, [(h ° ̃) ∩ ̃ (h ° ̃ ° h )]( ) = (h ° ̃)( ) ∩ (h ° ̃ ° h )( ),

S S S S S S

= [hS( ) ∪ hS( )] ∩ [hS( ) ∪ hS( )],

= hS( ) ∪ hS( ),

⊈ hS( ).

Hence, hS 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 fS is an SI-almost left BQ-ideal. Hence, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. We need to show that (S ° f ) ∩ ̃ f ≠ ∅ , for all x ∈ S.

x S S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ⊆ ̃ (S ° f ) ∩ ̃ f .

x S S y S S x S S

Since [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (S ° f ) ∩ ̃ f ≠ ∅ . Hence, f is an SI-almost left

x S S y S S S x S S S S

ideal.

Theorem 4. Every SI-almost (Left/right) BQ-ideal is an SI-almost bi-ideal.

Proof: Assume that f is an SI-almost left BQ-ideal. Hence, [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S.

S S x S y S S S

We need to show that (f ° S ° f ) ∩ ̃ f ≠ ∅ , for all x, y ∈ S. [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ⊆ ̃ (f ° S ° f ) ∩ ̃ f

S x S S S S x S y S S S x S S

Since [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (f ° S ° f ) ∩ ̃ f ≠ ∅ . Hence, f is an SI-almost bi-

S x S y S S S S x S S S S

ideal.

The following example shows that the converse of Theorem 4 is not true in general:

Example 3. Consider the soft set gS in Example 1. Here, gS is an SI-almost bi-ideal, that is, (g ° S ° g ) ∩ ̃ g ≠ ∅ , for all x ∈ S. Let's first show that (g ° S ° g ) ∩ ̃ g ≠ ∅ :

S x S S S S S S S [(g ° S ° g ) ∩ ̃ g ]( ) = (g ° S ° g )( ) ∩ g ( ),

S S S S S S

= [(gS( ) ∩ (S ° gS)( )) ∪ (gS( ) ∩ (S ° gS)( ))].

∩ gS( ) = [gS( ) ∪ gS( )] ∩ gS( ), = gS( ).

[(g ° S ° g ) ∩ ̃ g ]( ) = (g ° S ° g )( ) ∩ g ( ),

S S S S S S

= [(gS( ) ∩ (S ° gS)( )) ∪ (gS( ) ∩ (S ° gS)( ))] ∩ gS( ), = [gS( ) ∩ gS( )] ∩ gS( ), = gS( ) ∩ gS( ).

Consequently, (g ° S ° g ) ∩ ̃ g = {( , {x, y}), ( , ∅)} ≠ ∅ .

S S S S

Similarly, (g ° S ° g ) ∩ ̃ g = {( , ∅), ( , {e, yx})} ≠ ∅ .

S S S S

Thus, gS is an SI-almost bi-ideal. However, it is clear that gS is not an SI-almost (Left/right) BQ-ideal, as seen in Example 1.

Proposition 3. Let fS be an idempotent soft set. If fS is an SI-almost (Left/right) BQ-ideal, then fS is an SI-almost subsemigroup.

Proof: Assume that fS is an SI-almost left BQ-ideal such that fS is an idempotent, then fS ° fS = fS and [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. We need to show that f is an SI-almost subsemigroup, that

x S S y S S S S

is (f ° f ) ∩ ̃ f ≠ ∅ .

S S S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f = [(S ° f ) ∩ ̃ (f ° S ° f ) ∩ ̃ f ] ∩ ̃ f

x S S y S S x S S y S S S

= [(S ° f ) ∩ ̃ (f ° S ° f ) ∩ ̃ (f ° f )] ∩ ̃ f

x S S y S S S S

⊆ ̃ (f ° f ) ∩ ̃ f ,

S S S

Since [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (f ° f ) ∩ ̃ f ≠ ∅ . Thus, f is an SI-almost

x S S y S S S S S S S S

subsemigroup.

Proposition 4. Let fS be an idempotent soft set. If fS is an SI-almost left (Resp. right) BQ-ideal, then fS is an SI-almost left (Resp. right) weak interior ideal.

Proof: Assume that fS is an SI-almost left BQ-ideal such that fS is an idempotent, then fS ° fS = fS and [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. We need to show that f is an SI-almost left weak interior

x S S y S S S S

ideal, that is S ° f ° f ∩ ̃ f ≠ ∅ , for all x ∈ S.

x S S S S

[(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ⊆ ̃ (S ° f ) ∩ ̃ f = (S ° f ° f ) ∩ ̃ f ,

x S S y S S x S S x S S S

Since [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (S ° f ° f ) ∩ ̃ f ≠ ∅ . Thus, f is an SI-almost left

x S S y S S S x S S S S S

weak interior ideal.

Proposition 5. Let fS be an idempotent soft set. If fS is an SI-almost left (Right) BQ-ideal, then fS is an SI-almost tri-ideal.

Proof: Assume that fS is an idempotent SI-almost left BQ-ideal such that fS ° fS = fS and [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. We need to show that f is an SI-almost tri-ideal, that is

S x S y S S S S

(f ° S ° f ° f ) ∩ ̃ f ≠ ∅ and (f ° f ° S ° f ) ∩ ̃ f ≠ ∅ , for all x ∈ S.

S x S S S S S S x S S S

[(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ⊆ ̃ (f ° S ° f ) ∩ ̃ f = (f ° S ° f ° f ) ∩ ̃ f .

S x S y S S S x S S S x S S S.

Since [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (f ° S ° f ° f ) ∩ ̃ f ≠ ∅ . Hence, f is an SI-almost

S x S y S S S S x S S S S S

left tri-ideal.

[(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ⊆ ̃ (f ° S ° f ) ∩ ̃ f = (f ° f ° S ° f ) ∩ ̃ f .

S x S y S S S x S S S S x S S

Since [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (f ° f ° S ° f ) ∩ ̃ f ≠ ∅ . Hence, f is an SI-almost

S x S y S S S S S x S S S S

right tri-ideal. Thus, fS is an SI-almost tri-ideal.

Proposition 6. Let fS be an idempotent soft set. If fS is an SI-almost (Left/right) BQ-ideal, then fS is an SI-almost tri-bi-ideal.

Proof: Assume that fS is an SI-almost left BQ-ideal such that fS is an idempotent, then fS ° fS = fS and [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S. We need to show that f is an SI-almost tri-bi-ideal, that is

S x S y S S S S

(f ° f ° S ° f ° f ) ∩ ̃ f ≠ ∅ , for all x ∈ S.

S S x S S S S

[(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ⊆ ̃ (f ° S ° f ) ∩ ̃ f = (f ° f ° S ° f ° f ) ∩ ̃ f .

S x S y S S S x S S S S x S S S

Since [(f ° S ° f ) ∩ ̃ (S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that (f ° f ° S ° f ° f ) ∩ ̃ f ≠ ∅ . Thus, f is an SI-

S x S y S S S S S x S S S S S

almost tri-bi-ideal.

Theorem 5. Let ⊆ ̃ ℎ . If f is an SI-almost left (Resp. right) BQ-ideal, then h is an SI-almost left (Resp. right) BQ-ideal.

Proof: Assume that f is an SI-almost left BQ-ideal. Hence, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S.

S x S S y S S S

We need to show that [(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h ≠ ∅ , for all x, y ∈ S, In fact, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ⊆ ̃ [(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h ≠ ∅ .

x S S y S S x S S y S S S

Since [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , it is obvious that [(S ° h ) ∩ ̃ (h ° S ° h )] ∩ ̃ h ≠ ∅ . This

x S S y S S S x S S y S S S

completes the proof.

Theorem 6. If f and h are SI-almost left (Resp. right) BQ-ideals, then f ∪ ̃ h is an SI-almost left (Resp.

S S S S

right) BQ-ideal.

Proof: Since f is an SI-almost left BQ-ideal and f ⊆ ̃ f ∪ ̃ h , f ∪ ̃ h is an SI-almost left BQ-ideal by Theorem

S S S S S S

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 f or h be an SI-almost left (Resp. right) BQ-ideal, then f ∪ ̃ h is an SI-almost left (Resp.

S S S S

right) BQ-ideal.

Here, note that if f and h are SI-almost left (Resp. right) BQ-ideals, then f ∩ ̃ h needs not be an SI-almost

S S S S

left (Resp. right) BQ-ideal.

Example 4. Consider the SI-almost left (Resp. right) BQ-ideal fS and hS in Example 1. Since, f ∩ ̃ h = {( , ∅), ( , ∅)} = ∅ .

S S S

f ∩ ̃ h 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 x ∈ S and Y be a nonempty subset of S. Then, Sx ° SY = SxY. If X is a nonempty subset of S and y ∈ S, then SX ° Sy = SXy [46].

Theorem 7. Let A be a nonempty subset of S. Then, A is an almost left (Resp. right) BQ-ideal if and only if SA, the soft characteristic function of A, is an SI-almost left (Resp. right) BQ-ideal.

Proof: Assume that ∅ ≠ A is an almost left BQ-ideal. Then,(xA ∩ AyA) ∩ A ≠ ∅, for all x, y ∈ S, and so there exist j ∈ S such that j ∈ (xA ∩ AzA) ∩ A. Since, ([(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S )(j) = ((S ∩ ̃ S ) ∩ ̃ S ) (j),

x A A y A A xA AyA A

= ((S ) ∩ ̃ S ) (j),

xA∩AyA A

= S(xA∩AyA)∩A(j), = U, ≠ ∅.

It follows that [(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S ≠ ∅ . Thus, S is an SI-almost left BQ-ideal.

x A A y A A S A

Conversely, assume that S is an SI-almost left BQ-ideal. Hence, we have [(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S ≠

A x A A y A A

∅S, for all x, y ∈ S. In order to show that A is an almost left BQ-ideal, we should prove that A ≠ ∅ and (xA ∩ AyA) ∩ A ≠ ∅, for all x, y ∈ S. A ≠ ∅ is obvious from the assumption. Now, ∅ ≠ [(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S

S x A A y A A,

⇒ ∃ ∈ S ; ([(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S )( ) ≠ ∅,

x A A y A A

⇒ ∃ ∈ S ; ((S ∩ ̃ S ) ∩ ̃ S ) ( ) ≠ ∅,

xA AyA A

⇒ ∃ ∈ S ; (S ∩ ̃ S )( ) ≠ ∅,

xA∩AyA A

⇒ ∃ ∈ S ; S(xA∩AyA)∩A( ) = U,

⇒ ∈ ( ∩ ) ∩ .

Hence, (xA ∩ AyA) ∩ A ≠ ∅. Consequently, A is an almost left BQ-ideal.

Lemma 2. Let f ∈ S (U). Then, f ⊆ ̃ S [45].

S S S supp(fS)

Theorem 8. If fS is an SI-almost left (Resp. right) BQ-ideal, then supp(fS) is an almost left (Resp. right) BQ-ideal.

Proof: Assume that f is an SI-almost left BQ-ideal. Thus, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , for all x, y ∈ S.

S x S S y S S S

In order to show that supp(fS) is an almost left BQ-ideal, by Theorem 8, it is enough to show that Ssupp(f ) is an SI-almost left BQ-ideal. By Lemma 2, [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ⊆ ̃ [(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S ,

x S S y S S x supp(fS) supp(fS) y supp(fS) supp(fS)

and [(S ° f ) ∩ ̃ (f ° S ° f )] ∩ ̃ f ≠ ∅ , it implies that [(S ° S ) ∩ ̃ (S ° S ° S )] ∩ ̃ S ≠ ∅ . Consequently, S is an SI-almost left BQ-

x supp(fS) supp(fS) y supp(fS) supp(fS) S supp(fS)

ideal and by Theorem 7, supp(fS) is an almost left BQ-ideal.

The following example shows that the converse of Theorem 8 is not true in general:

Example 5. We know that gS is not an SI-almost left BQ-ideal in Example 1. Since supp(gS) = { , }, [( { , }) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, [( { , }) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, [( { , }) ∩ ({ , } {ℯ, }{ , })] ∩ { , } = { , } ≠ ∅, [( { , }) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, supp(gS) is an almost left BQ-ideal. Similarly, gS is not an SI-almost left BQ-ideal, but since [({ , } ) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, [({ , } ) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, [({ , } ) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, [({ , } ) ∩ ({ , } { , })] ∩ { , } = { , } ≠ ∅, supp(gS) is an almost right BQ-ideal. That is to say, supp(gS) is an almost BQ-ideal, although gS is not an SI-almost BQ-ideal.

Definition 14. Let f and h be SI-almost left (Resp. right) BQ-ideals such that h ⊆ ̃ f . If supp(h ) =

S S S S S

supp(fS), then fS is called a minimal SI-almost BQ-ideal.

Theorem 9. Let A be a nonempty subset of S. Then, A is a minimal almost left (Resp. right) BQ-ideal if and only if SA, the soft characteristic function of A, is a minimal SI-almost left (Resp. right) BQ-ideal.

Proof: Assume that A is a minimal, almost left BQ-ideal. Thus, A is an almost left BQ-ideal, and so SA is an SI-almost left BQ-ideal by Theorem 7. Let f be an SI-almost left BQ-ideal such that f ⊆ ̃ S . By Theorem 8,

S S A

supp(fS) is an almost left BQ-ideal and by Note 1 and Corollary 1, supp(fS) ⊆ supp(SA) = A.

Since A is a minimal almost left BQ-ideal, supp(fS) = supp(SA) = A. Thus, SA is a minimal SI-almost left BQ-ideal by Definition 14.

Conversely, let SA be a minimal SI-almost left BQ-ideal. Thus, SA is an SI-almost left BQ-ideal, and A is an almost left BQ-ideal by Theorem 7. Let B be an almost left BQ-ideal such that B ⊆ A. By Theorem 7, SB is an SI-almost left BQ-ideal, and by Theorem 2 (i), S ⊆ ̃ S . Since S is a minimal SI-almost left BQ-ideal,

B A A

B = supp(SB) = supp(SA) = A.

by Corollary 1. Thus, A is a minimal, almost left BQ-ideal.

Definition 15. Let f , g , and h be any SI-almost left (Resp. right) BQ-ideals. If h ° g ⊆ ̃ f implies that

S S S S S S

h ⊆ ̃ f or g ⊆ ̃ f , then f is called an SI-prime almost left (Resp. right) BQ-ideal.

S S S S S

Definition 16. Let f and h be any SI-almost left (Resp. right) BQ-ideals. If h ° h ⊆ ̃ f implies that h ⊆ ̃ f ,

S S S S S S S

then fS is called an SI-semiprime almost left (Resp. right) BQ-ideal.

Definition 17. Let f , g , and h be any SI-almost left (Resp. right) BQ-ideals. If (h ° g ) ∩ ̃ (g ° h ) ⊆ ̃ f

S S S S S S S S

implies that h ⊆ ̃ f or g ⊆ ̃ f , then f is called an SI-strongly prime almost left (Resp. right) BQ-ideal.

S S S S S

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 SP, the soft characteristic function of P, is an SI-prime almost left (Resp. right) BQ-ideal, then P is a prime almost left (Resp. right) BQ-ideal, where ∅ ≠ P ⊆ S.

Proof: Assume that SP is an SI-prime almost left BQ-ideal. Thus, SP is an SI-almost left BQ-ideal, and thus, P is an almost left BQ-ideal by Theorem 7. Let A and B be almost left BQ-ideals such that AB ⊆ P. Thus, by Theorem 7, SA and SB are SI-almost left BQ-ideals, and by Theorem 2 (i) and (iii), S ° S = S ⊆ ̃ S .

A B AB P

Since S is an SI-prime almost left BQ-ideal and S ° S ⊆ ̃ S , it follows that S ⊆ ̃ S or S ⊆ ̃ S . Therefore,

P A B P A P B P

by Theorem 2 (i), A ⊆ P or B ⊆ P. Consequently, P is a prime almost left BQ-ideal.

Theorem 11. If SP, the soft characteristic function of P, is an SI-semiprime almost left (Resp. right) BQ-ideal, then P is a semiprime almost left (Resp. right) BQ-ideal, where ∅ ≠ P ⊆ S.

Proof: Assume that SP is an SI-semiprime almost left BQ-ideal. Thus, SP is an SI-almost left BQ-ideal, and thus, P is an almost left BQ-ideal by Theorem 7. Let A be an almost left BQ-ideal such that AA ⊆ P. Thus, by Theorem 7, SA is an SI-almost left BQ-ideal, and by Theorem 2 (i) and (ii), S ° S = S ⊆ ̃ S .

A A AA P

Since S is an SI-prime almost left BQ-ideal and S ° S ⊆ ̃ S , it follows that S ⊆ ̃ S . Therefore, by Theorem

P A A P A P

2 (i) A ⊆ P. Consequently, P is a semiprime almost left BQ-ideal.

Theorem 12. If SP, the soft characteristic function of P, is an SI-strongly prime almost left (Resp. right) BQ-ideal, then P is a strongly prime almost left (Resp. right) BQ-ideal, where ∅ ≠ ⊆ .

Proof: Assume that SP is an SI-strongly prime almost left BQ-ideal. Thus, SP is an SI-almost left BQ-ideal, and thus, P is an almost left BQ-ideal by Theorem 7. Let A and B be almost left BQ-ideals such that AB ∩ BA ⊆ P. Thus, by Theorem 7, SA and SB are SI-almost left BQ-ideals, and by Theorem 2, (S ° S ) ∩ ̃ (S ° S ) = S ∩ ̃ S = S ⊆ ̃ S ,

A B B A AB BA AB∩BA P

Since S is an SI-strongly prime almost left BQ-ideal and (S ° S ) ∩ ̃ (S ° S ) ⊆ ̃ S , it follows that S ⊆ ̃ S or

P A B B A P A P

S ⊆ ̃ S . Thus, by Theorem 2 (i), A ⊆ P or B ⊆ P. Therefore, P is a strongly prime, almost left BQ-ideal.

The concepts of "soft intersection almost bi-quasi ideal" and "soft intersection weakly almost bi-quasi ideal" 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 intersection almost bi-ideal of a semigroup; the converses are not true in general with counterexamples. Additionally, it was shown 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. We obtained the relation between soft intersection almost bi-quasi ideal 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 A is almost bi-quasi ideal then its soft characteristic function is soft intersection almost bi-quasi ideal, and vice versa. Additionally, we investigated that, unlike soft intersection operation, soft union operation can form a semigroup with the collection of almost bi-quasi ideals 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.

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