# Interior GE-Algebras

Authors: Jeong-Gon Lee; Ravikumar Bandaru; Kul Hur; Young Bae Jun
Journal: Journal of Mathematics
 (2021)

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## Abstract

The concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras are introduced, and their relations and properties are investigated. Many examples are given to support these concepts. A semigroup is formed using the set of interior GE-algebras. An example is given that the set of interior GE-algebras is not a GE-algebra. It is clear that ifX is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra X,f is transitive (resp., commutative, belligerent, and left exchangeable), but examples are given to show that the converse is not true in general. An interior GE-algebra is constructed using a bordered interior GE-algebra with certain conditions, and an example is given to explain this.


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## Body

## 1. Introduction

In 1966, Imai and Iséki introduced BCK-algebras (see [1]) as the algebraic semantics for a nonclassical logic possessing only implication. Since then, the generalized concepts of BCK-algebras have been studied by various scholars. Kim and Kim introduced the notion of a BE-algebra as a generalization of a dual BCK-algebra (see [2]). Hilbert algebras were introduced by Henkin and Skolem in the fifties for investigations in intuitionistic and other nonclassical logics. Diego proved that Hilbert algebras form a variety which is locally finite (see [3]). Rezaei et al. discussed relations between Hilbert algebras and BE-algebras (see [4]). The generalization process in the study of algebraic structures is also an important area of study. As a generalization of Hilbert algebras, Bandaru et al. introduced the notion of GE-algebras and investigated several properties (see [5–8]).The notion of the interior operator was introduced by Vorster [9] in an arbitrary category, and it was used in [10] to study the notions of connectedness and disconnectedness in topology. Interior algebras are a certain type of algebraic structures that encode the idea of the topological interior of a set and are a generalization of topological spaces defined by means of topological interior operators. Rachůnek and Svoboda [11] studied interior operators on bounded residuated lattices, and Svrcek [12] studied multiplicative interior operators on GMV-algebras.In this article, we apply the interior operator theory to GE-algebras. We introduce the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigate their relations and properties. We find and present many examples to illustrate these concepts. We use the set of interior GE-algebras to make up a semigroup. We give examples to show that the set of interior GE-algebras is not a GE-algebra. It is clear that ifX is a transitive (resp., commutative, belligerent and left exchangeable) GE-algebra, then the interior GE-algebra X,f is transitive (resp., commutative, belligerent and left exchangeable), but we give examples to show that its inverse is not established. We make up the internal GE-algebra using a bordered interior GE-algebra with certain conditions and give examples describing this.


## 2. Preliminaries

Definition 1.
(see [5]). By a GE-algebra, we mean a nonempty set X with constant 1 and binary operation ∗ satisfying the following axioms:(GE1)u∗u=1,(GE2)1∗u=u,(GE3)u∗v∗w=u∗v∗u∗w,
for allu,v,w∈X.In a GE-algebraX, a binary relation “≤” is defined by(1)∀x,y∈X,x≤y⟺x∗y=1.Definition 2.
(see [5, 6, 8]). A GE-algebra X is said to be(i)
transitive if it satisfies(2)∀x,y,z∈X,x∗y≤z∗x∗z∗y.(ii)
commutative if it satisfies(3)∀x,y∈X,x∗y∗y=y∗x∗x.(iii)
left exchangeable if it satisfies(4)∀x,y,z∈X,x∗y∗z=y∗x∗z.(iv)
belligerent if it satisfies(5)∀x,y,z∈X,x∗y∗z=x∗y∗x∗z.(v)
antisymmetric if the binary relation “≤” is antisymmetric.Proposition 1 (see [5]).
Every GE-algebraX satisfies the following items:(6)∀u∈X,u∗1=1,∀u,v∈X,u∗u∗v=u∗v,∀u,v∈X,u≤v∗u,∀u,v,w∈X,u∗v∗w≤v∗u∗w,∀u∈X,1≤u⟹u=1,∀u,v∈X,u≤v∗u∗u,∀u,v∈X,u≤u∗v∗v,∀u,v,w∈X,u≤v∗w⟺v≤u∗w.
IfX is transitive, then(7)∀u,v,w∈X,u≤v⟹w∗u≤w∗v,v∗w≤u∗w,(8)∀u,v,w∈X,u∗v≤v∗w∗u∗w.Lemma 1 (see [5]).
In a GE-algebraX, the following facts are equivalent to each other:(9)∀x,y,z∈X,x∗y≤z∗x∗z∗y,∀x,y,z∈X,x∗y≤y∗z∗x∗z.


## 3. Interior GE-Algebras

Definition 3.
By an interior GE-algebra, we mean a pairX,f in which X is a GE-algebra and f:X⟶X is a mapping such that(10)∀x∈X,x≤fx,(11)∀x∈X,f∘fx=fx,(12)∀x,y∈X,x≤y⟹fx≤fy.Example 1.
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ which is given in the following table:(13)∗1abcd11abcda111ccb111ddc1ab11d1aa11.
Then, it is routine to verify thatX,f is an interior GE-algebra, where(14)f:X⟶X,x⟼1,ifx∈1,c,d,a,ifx∈a,b.It is clear that ifX is a GE-algebra, then X,idX and X,1˜ are interior GE-algebras, where idX:X⟶X,x⟼x and 1˜:X⟶X,x⟼1.In the following example, we know that there is a constant mapc˜:X⟶X,x⟼c, where c≠1∈X, on a GE-algebra X such that X,c˜ is not an interior GE-algebra.Example 2.
Consider a GE-algebraX,∗,1, where X=1,a,b,c,d and ∗ is a binary operation on X, which is given in the following table:(15)∗1abcd11abcda11b11b1d11dc1a11ad11bc1.
If we take a constant mappingc˜:X⟶X,x⟼c, then d≰c=c˜d. Hence, X,c˜ is not an interior GE-algebra.LetIntX be the set of all interior GE-algebras. For every X,f,X,g∈IntX, we define(16)X,f=X,g⟺f=g,i.e,fx=gx,for allx∈X,X,f∘X,g≔X,f∘g,X,f⊛X,g≔X,f⊛g,where f∘g:X⟶X,x⟼fgx and f⊛g:X⟶X,x⟼fx∗gx.LetX,f,X,g∈IntX. The following example shows that the composition X,f∘X,g=X,f∘g of X,f and X,g may not be an interior GE-algebra, and X,f∘g≠X,g∘f.Example 3.
LetX=1,a,b,c,d be a set with the binary operation ∗ given in the following table:(17)∗1abcd11abcda111ccb111ddc11b11d11111.
Then,X is a GE-algebra. Define two mappings:(18)f:X⟶X,x⟼1,ifx=1,a,ifx∈a,b,d,ifx∈c,d,g:X⟶X,x⟼1,ifx=1,a,ifx∈a,c,b,ifx∈b,d.
Then,X,f and X,g are interior GE-algebras, and the composition g∘f of f and g is calculated as follows:(19)g∘f:X⟶X,x⟼1,ifx=1,a,ifx∈a,b,b,ifx∈c,d.
Sincec∗g∘fc=c∗gfc=c∗gd=c∗b=b≠1, the composition X,g∘X,f of X,f and X,g is not an interior GE-algebra. Also, X,f∘g≠X,g∘f since(20)f∘gc=fgc=fa=a≠b=gd=gfc=g∘fc.We consider the following condition:(21)X,f∘g=X,g∘f,for X,f,X,g∈IntX.Denote bycIntX the set of all interior GE-algebras satisfying condition (21).Theorem 1.
IfX is a GE-algebra, then cIntX,∘ is a semigroup.Proof.
It is sufficient to show thatcIntX is closed under ∘. Let X,f,X,g∈cIntX. Using (10), we have x≤gx≤fgx=f∘gx for all x∈X, and so, f∘g satisfies condition (10). Also,(22)f∘g∘f∘gx=f∘f∘g∘gx=f∘fg∘gx=f∘fgx=fgx=f∘gx,for all x∈X, which shows that f∘g satisfies condition (11). For every x,y∈X, if x≤y, then gx≤gy, and so, f∘gx=fgx≤fgy=f∘gy. This shows that X,f∘X,g=X,f∘g is an interior GE-algebra, that is, cIntX is closed under ∘. Therefore, cIntX,∘ is a semigroup.The following example describes Theorem1.Table 1
Self-mapsfi, i=1,2.
x1abf1x11bf2x1a1Example 4.
(1)
Consider a GE-algebraX,∗,1, where X=1,a,b and ∗ is a binary operation on X, which is given in the following Cayley table:(23)∗1ab11aba11bb1a1.The set of all interior GE-algebras isIntX=X,1˜,X,idX,X,f1,X,f2, where the self-maps f1 and f2 are given by Table 1.We can checkcIntX=IntX by the following Cayley table:(24)∘1˜idXf1f21˜1˜1˜1˜1˜idX1˜idXf1f2f11˜f1f11˜f21˜f21˜f2.AndcIntX,∘ is a semigroup.(2)
Consider a GE-algebraX=1,a,b,c,d with the binary operation which is given in the following table:(25)∗1abcd11abcda111ddb111ccc11b11d11b11.The set of all interior GE-algebrasIntX consists of(26)IntX=X,1˜,X,idX,X,fi,i=1,2,…,10,in which eachfi, i=1,2,…,10, is given in Table 2.The operation “°” in IntX is calculated as follows:(27)∘1˜idXf1f2f3f4f5f6f7f8f9f101˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜idX1˜idXf1f2f3f4f5f6f7f8f9f10f11˜f1f1f1f1f1f1f1f1f1f1f1f21˜f2f1f2f2f4f1f6f6f6f2f2f31˜f3f1f2f3f4f1f6f7f8f2f4f41˜f4f1f4f4f4f1f8f8f8f4f4f51˜f5f1f1f1f1f5f11f11f11f5f5f61˜f61˜f6f6f61˜f6f6f6f6f6f71˜f71˜f6f7f81˜f6f7f8f6f8f81˜f81˜f8f8f81˜f8f8f8f8f8f91˜f9f1f2f2f2f5f6f6f6f9f9f101˜f10f1f4f4f4f5f8f8f8f10f10.in whichX,f11, which is not contained in IntX, is given as follows:We know thatintX is not closed under the operation “°,” and cIntX,° is a semigroup, where cIntX=1˜,idX,f1,f2,f3,f4,f5.LetX,f,X,g∈IntX. The following example shows that X,f⊛X,g may not be an interior GE-algebra, and X,f⊛g≠X,g⊛f.Table 2
Self-mapsfi, i=1,2,…,10.
x1abcdf1x1aaaaf2x1aaccf3x1aacdf4x1aaddf5x1abaaf6x111ccf7x111cdf8x111ddf9x1abccf10x1abddx1abcdf11x111aaExample 5.
ConsiderIntX in Example 4 (2). Then, the operation “⊛” in IntX is calculated as follows:(28)⊛1˜idXf1f2f3f4f5f6f7f8f9f101˜1˜idXf1f2f3f4f5f6f7f8f9f10idX1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜f11˜f81˜f8f8f81˜f8f8f8f8f8f21˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜f31˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜f41˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜f51˜f81˜f8f8f81˜ff8f8f8f8f8f61˜f6idf12f12f12f12f6id1˜1˜1˜f6idf6idf71˜f7idf12f12f12f12f7id1˜1˜1˜f7idf7idf81˜f8idf12f12f12f12f8id1˜1˜1˜f8idf8idf91˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜f101˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜1˜,in which f6id,f7id,f8id,f12∈IntX, and they are calculated as follows:
We also know thatX,f⊛g≠X,g⊛f; for example, f1⊛f7=f8≠f12=f7⊛f1.x1abcdf6idx1ab11f7idx1ab11f8idx1ab11f12x1aa11Example5 generally shows that IntX,⊛,1˜ cannot be a GE-algebra. However, the following example shows that IntX,⊛,1˜ becomes a GE-algebra sometimes.Example 6.
ConsiderIntX=X,1˜,X,idX,X,f1,X,f2 in Example 4 (1). Then, the operation “⊛” in IntX is calculated as follows:(29)⊛1˜idXf1f21˜1˜idXf1f2idX1˜1˜1˜1˜f11˜f21˜f2f21˜f1f11˜.
It is routine to verify thatIntX,⊛,1˜ is a GE-algebra.For everyX,f,X,g∈IntX, we define(30)X,f≪X,g⟺∀x∈X,gx≤fx.For everyX,f∈IntX, the sets(31)ℐf≔x∈X|fx=x,kerf≔x∈X|fx=1are called the identity part and the kernel of f, respectively.Proposition 2.
LetX be an antisymmetric and transitive GE-algebra. For every X,f,X,g∈IntX, we have(i)
X,f≪X,g⟺X,f∘X,g=X,f(ii)
X,f=X,g⟺ℐf=ℐgProof.
(i)
IfX,f≪X,g, then gx≤fx for all x∈X, and so, f∘gx≤f∘fx=fx for all x∈X. Also, x≤gx=g∘gx≤f∘gx which implies that fx≤f∘f∘gx=f∘fgx=f∘gx for all x∈X. Hence, f∘gx=fx for all x∈X, and therefore, X,f∘g=X,f. Conversely, assume that X,f∘g=X,f. Then, gx≤f∘gx=fx for all x∈X. Thus, X,f≪X,g.(ii)
It is clear that ifX,f=X,g, then ℐf=ℐg. Suppose that ℐf=ℐg. Condition (11) induces fx∈ℐf=ℐg, and so, gfx=fx for all x∈X. Hence, g∘f=f. Similarly, f∘g=g. Using (10) and (12), we have gx≤g∘fx=fx and fx≤f∘gx=gx. Thus, fx=gx for all x∈X, and therefore, X,f=X,g.Lemma 2 (see [5]).
Every GE-algebraX satisfies(i)x≤y∗x,(ii)x≤x∗y∗y,(iii)x≤y∗x∗x,(iv)x≤x∗y∗x,(v)x≤y∗y∗x,(vi)x∗y∗z≤y∗x∗z,
for allx,y,z∈X.Proposition 3.
IfX,f is an interior GE-algebra, then(i)
f1=1(ii)
∀x,y∈X,fx≤fy∗x(iii)
∀x,y∈X,fx≤fx∗y∗y(iv)
∀x,y∈X,fx≤fy∗x∗x(v)
∀x,y∈X,fx≤fx∗y∗x(vi)
∀x,y∈X,fx≤fy∗y∗x(vii)
∀x,y,z∈X,fx∗y∗z≤fy∗x∗zProof.
(i) is straightforward, and (ii)–(vii) follow from (12) and Lemma 2.Question 1.
IfX,f is an interior GE-algebra, will the next items be established?(32)∀x,y,z∈X,fx∗y≤fz∗x∗z∗y,(33)∀x,y∈X,fx∗y∗y=fy∗x∗x,(34)∀x,y,z∈X,fx∗y∗z=fx∗y∗x∗z,(35)∀x,y,z∈X,fx∗y∗z=fy∗x∗z.The following example shows that the answer to the above question is negative.Example 7.
(1)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(36)∗1abcd11abcda11b11b1a111c1ab11d111c1.Definef:X⟶X by(37)fx=1,ifx=1,b,ifx=b,d,ifx∈a,c,d.Then,X,f is an interior GE-algebra, but X,f does not satisfy (32) and (33) since(38)fb∗c∗fd∗b∗d∗c=f1∗f1∗c=1∗fc=1∗d=d≠1,fb∗c∗c=f1∗c=fc=d≠1=f1=fb∗b=fc∗b∗b.(2) Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(39)∗1abcd11abcda11cc1b1d11dc11111d11cc1.Definef:X⟶X by(40)fx=1,ifx=1,c,ifx∈b,c,d,ifx∈a,d.Then,X,f is an interior GE-algebra, but X,f does not satisfy (34) since(41)fb∗c∗d=fb∗1=f1=1≠d=fd=f1∗d=fb∗c∗b∗d.(3)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(42)∗1abcd11abcda111ccb111ddc1aa11d1aa11.Definef:X⟶X by(43)fx=1,ifx=1,a,ifx∈a,b,c,ifx=c,d,ifx=d.Then,X,f is an interior GE-algebra, but X,f does not satisfy (35) since(44)fa∗b∗c=fa∗d=fc=c≠d=fd=fb∗c=fb∗a∗c.Definition 4.
An interior GE-algebraX,f is said to be transitive (resp., commutative, belligerent, and left exchangeable) if it satisfies (32) (resp., (33), (34), and (35)).Example 8.
(1)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(45)∗1abcd11abcda111cdb1a1ddc11b11d11b11.Definef:X⟶X by(46)fx=1,ifx∈1,a,b,c,ifx=c,d.Then,X,f is a transitive interior GE-algebra.(2)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(47)∗1abcd11abcda11abcb1a1c1c1ab11d111c1.Definef:X⟶X by(48)fx=1,ifx∈1,a,b,d,c,ifx=c.Then,X,f is a commutative interior GE-algebra.(3)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(49)∗1abcd11abcda11bcdb111cdc1111dd1abc1.Definef:X⟶X by(50)fx=1,ifx=1,a,ifx=a,b,ifx∈b,c,d,ifx=d.Then,X,f is a belligerent interior GE-algebra.(4)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(51)∗1abcd11abcda1111db1c1c1c1bb11d1bb11.Definef:X⟶X by(52)fx=1,ifx=1,b,ifx∈a,b,d,ifx∈c,d.Then,X,f is a left exchangeable interior GE-algebra.It is clear that ifX is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra X,f is transitive (resp., commutative, belligerent, and left exchangeable), but the converse is not true in general as seen in the following example.Example 9.
(1)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(53)∗1abcd11abcda111cdb1111dc1ab1dd1aac1.Definef:X⟶X by(54)fx=1,ifx∈1,c,d,a,ifx∈a,b.Then,X,f is a transitive interior GE-algebra, but X is not transitive GE-algebra since(55)b∗c∗d∗b∗d∗c=1∗a∗c=a∗c=c≠1.(2)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(56)∗1abcd11abcda111cdb111cdc1bb1dd1bb11.Definef:X⟶X by(57)fx=1,ifx∈1,c,b,ifx∈a,b,d,ifx=d.Then,X,f is a commutative interior GE-algebra, but X is not commutative GE-algebra since(58)a∗b∗b=1∗b=b≠a=1∗a=b∗a∗a.(3)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(59)∗1abcd11abcda11cc1b1a11dc1a11ad11cc1.
Definef:X⟶X by(60)fx=1,ifx=1,d,ifx∈a,d,c,ifx∈b,c.Then,X,f is a belligerent interior GE-algebra, but X is not belligerent GE-algebra since(61)b∗c∗d=b∗a=a≠d=1∗d=b∗c∗b∗d.(4)
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(62)∗1abcd11abcda11cc1b1d11dc1a11ad11cc1.Definef:X⟶X by(63)fx=1,ifx=1,c,ifx∈b,c,d,ifx∈a,d.Then,X,f is a left exchangeable interior GE-algebra, but X is not left exchangeable GE-algebra since(64)b∗c∗d=b∗a=d≠a=c∗d=c∗b∗d.The following example shows that any interior GE-algebraX,f does not satisfy the following:(65)∀x,y∈X,fx∗y≤x∗fy,(66)∀x,y∈X,fx∗y≤fx∗y.Example 10.
Consider a GE-algebraX=1,a,b,c,d with the binary operation ∗ given in the following Cayley table:(67)∗1abcd11abcda11bc1b111c1c1db1dd11b11.
Definef:X⟶X by(68)fx=1,ifx=1,a,ifx=a,c,ifx=c,d,ifx∈b,d.
Then,X,f is an interior GE-algebra, but X does not satisfy (65) and (66) since(69)fb∗c∗b∗fc=d∗c∗b∗c=1∗c=c≠1,fb∗c∗fb∗c=d∗c∗fc=1∗c=c≠1.Proposition 4.
Every transitive interior GE-algebraX,f satisfies (65) and (66).Proof.
Letx,y∈X. Using (7) and (10) induces fx∗y≤fx∗fy≤x∗fy which proves (65). Since x∗y≤fx∗y and x≤fx, it follows from (7) that fx∗y≤x∗y≤fx∗y. Hence, (66) is valid.Definition 5.
(see [7]). If a GE-algebra X has a special element, say 0, which satisfies 0≤x for all x∈X, we call X the bordered GE-algebra.Definition 6.
(see [7]). By a duplex bordered element in a bordered GE-algebra X, we mean an element x of X which satisfies x00=x.The set of all duplex bordered elements of a bordered GE-algebraX is denoted by 02X and is called the duplex bordered set of X. It is clear that 0,1∈02X.Definition 7.
(see [7]). A bordered GE-algebra X is said to be duplex if every element of X is a duplex bordered element, that is, X=02X.Definition 8.
By a bordered interior GE-algebra, we mean an interior GE-algebraX,f in which X is a bordered GE-algebra.Example 11.
Consider a bordered GE-algebraX=0,1,a,b,c with the binary operation ∗ given in the following Cayley table:(70)∗01abc011111101abca111b1ba1a11c010b1.
Definef:X⟶X by(71)fx=0,ifx∈0,a,1,ifx=1,c,ifx∈b,c.
It is routine to verify thatX,f is a bordered interior GE-algebra.Proposition 5.
In a bordered interior GE-algebraX,f in which X is transitive, we have(72)∀x∈X,fx0≤fx0,∀x,y∈X,x∗y≤fy0∗x0.Proof.
If we takey=0 in (66), then fx0=fx∗0≤fx∗0=fx0. Taking u=x, v=y, and w=0 in (8) induces x∗y≤y0∗x0, and so, x∗y≤fx∗y≤fy0∗x0 by (10) and (12).Lemma 3 (see [7]).
The duplex bordered set02X of a transitive and antisymmetric bordered GE-algebra X is closed under the binary operation ∗ in X, that is, it is a GE-subalgebra of X and is also bordered.Theorem 2.
LetX,f be a transitive and antisymmetric bordered interior GE-algebra. Then, 02X,f˜ is an interior GE-algebra, where(73)f˜:02X⟶02X,x⟼fx00.Proof.
Letx∈02X. Then, x=x00≤fx00=f˜x by (7), and(74)f˜∘f˜x=f˜fx00=fx0000=fx00=f˜x.
Letx,y∈02X be such that x≤y. Then, fx≤fy by (12), and thus, f˜x=fx00≤fy00=f˜y. This completes the proof.The following example describes Theorem2.Example 12.
Consider a GE-algebraX=0,1,a,b,c,d,e with the binary operation ∗ given in the following Cayley table:(75)∗01abcde01111111101abcdeac111cc1b01a1cdeca1a11eeda1a1111e01a1cc1.
Definef:X⟶X by(76)fx=0ifx=01ifx=1,bifx∈a,b,e,cifx∈c,d.
Then,X,f is a transitive and antisymmetric bordered interior GE-algebra, and 02X,∗,1 is a GE-algebra, where 02X=0,1,a,c. We know that f˜:02X⟶02X is calculated as follows:(77)f˜x=0,ifx=0,1,ifx∈1,a,c,ifx=c,and it is routine to observe that 02X,f˜ is an interior GE-algebra.


## 4. Conclusions

We have introduced the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigated their relations and properties. We have found and presented many examples to illustrate these concepts. We have formed a semigroup using the set of interior GE-algebras. We have provided examples to show that the set of interior GE-algebras is not a GE-algebra. It is clear that ifX is a transitive (resp., commutative, belligerent, and left exchangeable) GE-algebra, then the interior GE-algebra X,f is transitive (resp., commutative, belligerent, and left exchangeable), but we have considered examples to show that its inverse is not established. We have provided examples of how to construct and explain interior GE-algebra using a bordered interior GE-algebra under certain conditions. In the future work, we will use the idea and results given in this paper to study other (hyper) algebraic structures, for example, (hyper) hoop, (hyper) BCH-algebra, (hyper) equality algebra, and (hyper) MV-algebra.



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