The concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored.
1. Introduction
Recently soft set theory has emerged as a new mathematical tool to deal with uncertainty. Due to its applications in various fields of study researchers and practitioners are showing keen interest in it. As enough number of parameters is available here, so it is free from the difficulties associated with other contemporary theories dealing with uncertainty. Prior to soft set theory, probability theory, fuzzy set theory, rough set theory, and interval mathematics were common mathematical tools for dealing with uncertainties, but all these theories have their own difficulties. These difficulties may be due to lack of parametrization tools [1, 2]. To overcome these difficulties, Molodtsov [2] introduced the concept of soft sets. A detailed overview of these difficulties can be seen in [1, 2]. As a new mathematical tool for dealing with uncertainties, Molodtsov has pointed out several directions for the applications of soft sets. Theoretical development of soft sets is due to contributions from many researchers. However in this regard initial work is done by Maji et al. in [1]. Later Ali et al. [3] introduced several new operations in soft set theory.
At present, work on the soft set theory is progressing rapidly. Maji et al. [4] described the application of soft set theory in decision making problems. Aktaş and Çağman studied the concept of soft groups and derived their basic properties [5]. Chen et al. [6] proposed parametrization reduction of soft sets, and then Kong et al. [7] presented the normal parametrization reduction of soft sets. Feng and his colleagues studied roughness in soft sets [8, 9]. Relationship between soft sets, fuzzy sets, and rough sets is investigated in [10]. Park et al. [11] worked on notions of soft WS-algebras, soft subalgebras, and soft deductive system. Jun and Park [12] presented the notions of soft ideals, idealistic soft, and idealistic soft BCI/BCK-algebras. Further applications of soft sets can be seen in [13–25].
The study of BCI/BCK-algebras was initiated by Imai and Iseki [26] as the generalization of concept of set theoretic difference and propositional calculus. For the general development of BCI/BCK-algebras, the ideal theory and its fuzzification play an important role. Jun et al. [27–30] studied fuzzy trends of several notions in BCI/BCK-algebras. Application of soft sets in BCI/BCK is given in [12, 31].
Translations play a vital role in reducing the complexity of a problem. In geometry it is a common practice to translate a system to some new position to study its properties. In linear algebra translations help find solution to many practical problems. In this paper idea of translations is being extended to soft BCI/BCK algebras.
This paper is arranged as follows: in Section 2, some basic notions about BCI/BCK-algebra and soft sets are given. These notions are required in the later sections. Concept of translation is introduced in Section 3 and some properties of it are discussed here. Section 4 is devoted for the study of soft ideal translation in BCI/BCK-algera. In Section 5, concept of ideal extension is introduced and some of its properties are studied.
2. Preliminaries
First of all some basic concepts about BCI/BCK-algebra are given. For a comprehensive study on BCI/BCK-algebras [32] is a very nice monograph by Meng and Jun. Then some notions about soft sets are presented here as well.
An algebra (X,*,0) is called a BCI-algebra if it satisfies the following conditions:
(∀x,y,z∈X)(((x*y)*(x*z))*(z*y)=0),
(∀x,y∈X)((x*(x*y))*y=0),
(∀x∈X)(x*x=0),
(∀x,y∈X)(x*y=0,y*x=0⇒x=y).
If a BCI-algebra satisfies the following identity:
(∀x∈X)(0*x=0),
then X is called a BCK-algebra. Any BCK-algebra satisfies the following axioms:
(∀x∈X)(x*0=x),
(∀x,y,z∈X)(x*y=0⇒(x*z)*(y*z)=0,(z*y)*(z*x)=0),
(∀x,y,z∈X)((x*y)*z=(x*z)*y),
(∀x,y,z∈X)(((x*z)*(y*z))*(x*y)=0).
A subset S of a BCI/BCK-algebra X is called a subalgebra of X if x*y∈S, for all x,y∈S.
A subset A of a BCI/BCK-algebra X is called an ideal of X, denoted by A⊲X, if it satisfies:
0∈A,
(∀x,y∈X)(x*y∈A,y∈A⇒x∈A).
Now we recall some basic notions in soft set theory. Let U be a universe and E be a set of parameters. Let P(U) denote the power set of U and let A, B be nonempty subsets of E.
Definition 1 (see [<xref ref-type="bibr" rid="B29">2</xref>]).
A pair (F,A) is called a soft set over U, where F is a mapping given by F:A→P(U).
Definition 2 (see [<xref ref-type="bibr" rid="B2">3</xref>]).
Let U be a universe, let E be the set of parameters, and let A⊆E.
(F,A) is called a relative null soft set (with respect to the parameters set A), denoted by ∅A, if F(a)=∅, for all a∈A.
(G,A) is called a relative whole soft set (with respect to the parameters set A), denoted by UA, if G(e)=U, for all e∈A.
Definition 3 (see [<xref ref-type="bibr" rid="B2">3</xref>]).
The complement of a soft set (F,A) is denoted by (F,A)c and is defined by (F,A)c=(Fc,A), where Fc:A→P(U) is a mapping given by Fc(a)=U-F(a), ∀a∈A. Clearly, ((F,A)c)c=(F,A).
Definition 4 (see [<xref ref-type="bibr" rid="B9">8</xref>]).
A soft set (F,A) over U is called a full soft set if ⋃a∈AF(a)=U.
3. Soft Translations of Soft Subalgebras
Here notion of translations in soft BCI/BCK-algebra is initiated. Concept of soft extensions is introduced here also.
Let FA:X→P(X) be set valued map defined as
(1)FA(x)≠∅ifx∈A,
where A⊆X. Then FA also denotes a soft set over a BCI/BCK algebra X. From here onward a soft set will be denoted by symbols like FA, unless stated otherwise.
A soft set FA over a BCI/BCK-algebra X is called a soft subalgebra of X if it satisfies
(2)(∀x,y∈X)(FA(x*y)⊇FA(x)∩FA(y)).
In what follows X=(X,*,0) denote a BCI/BCK-algebra, and for any soft set FA over X, we denote T:=X-∪{FA(x)∣x∈X} unless otherwise specified.
That is T=(⋃x∈XFA(x))c=⋂x∈XFAc(x).
It is easy to see that T∩FA(x)=∅ for all x∈X. If FA is a full soft set then T is an empty set. Therefore throughout this paper only those soft set are considered which are not full.
Definition 5.
Let FA be a soft set over X and let U1⊆T. A mapping FU1T:X→P(X) is called a soft U1-translation of FA if, for all x∈X,
(3)FU1T(x)=FA(x)∪U1.
Lemma 6.
Let U1⊆T and FA be a soft set over X, then FA(x)∪U1⊇FA(y)∪U1 implies FA(x)⊇FA(y), for all x,y∈X.
Proof.
Since U1⊆T, FA(x)∩U1=∅ and FA(y)∩U1=∅. Let a∈FA(y) then a∈FA(y)∪U1⊆FA(x)∪U1 this implies a∈FA(x) or a∈U1 but a∉U1 because FA(y)∩U1=∅. So a∈FA(x) that is FA(x)⊇FA(y), for all x,y∈X.
Proposition 7.
Let FA be a soft subalgebra of X and U1⊆T. Then the soft U1-translation FU1T of FA is a soft subalgebra of X.
Proof.
Let x,y∈X. Then
(4)FU1T(x*y)=FA(x*y)∪U1⊇(FA(x)∩FA(y))∪U1=(FA(x)∪U1)∩(FA(y)∪U1)=(FU1T(x))∩(FU1T(y)).
Hence FU1T is a soft subalgebra of X.
Proposition 8.
Let FA be a soft set over X such that the U1-translation FU1T of FA is a soft subalgebra of X for some U1⊆T. Then FA is a soft subalgebra of X.
Proof.
Assume FU1T is a soft subalgebra of X for some U1⊆T. Let x,y∈X, we have
(5)FA(x*y)∪U1=FU1T(x*y)⊇FU1T(x)∩FU1T(y)=(FA(x)∪U1)∩(FA(y)∪U1)=(FA(x)∩(y))∪U1.
Now by Lemma 6 we have
(6)FA(x*y)⊇FA(x)∩FA(y),
for all x,y∈X. Hence FA is a soft subalgebra of X.
From Propositions 7 and 8 we have the following.
Theorem 9.
A soft set FA of X is a soft subalgebra of X if and only if U1-translation FU1T of FA is a soft subalgebra of X for some U1⊆T.
Definition 10.
Let FA and GB be two soft sets over X. If FA(x)⊆GB(x) for all x∈X, then we say that GB is a soft extension of FA.
Example 11.
Consider a BCI/BCK-algebra X={0,1,2,3} presented as follows:
(7)*012300000110112220233330
Define two soft sets FA and GB of X as in Table 1.
Here FA(0)⊆GB(0), FA(1)⊆GB(1), FA(2)⊆GB(2), and FA(3)⊆GB(3), which implies that GB is a soft extension of FA.
X
0
1
2
3
FA
{0}
{0,1}
{0,2}
{1,2}
GB
{0}
{0,1,2}
{0,2}
{0,1,2}
Next the concept of soft S-extension is being introduced.
Definition 12.
Let FA and GB be two soft sets over X. Then GB is called a soft S-extension of FA, if the following conditions hold:
GB is a soft extension of FA.
If FA is a soft subalgebra of X, then GB is a soft subalgebra of X.
As we know FU1T(x)⊇FA(x) for all x∈X. As a consequence of Definition 12 and Theorem 9, we have the following.
Theorem 13.
Let FA be a soft subalgebra of X and U1⊆T. Then the soft U1-translation FU1T of FA is a soft S-extension of FA.
The converse of Theorem 13 is not true in general as seen in the following example.
Example 14.
Consider a BCI/BCK-algebra X={0,1,2,3} given as follows:
(8)*012300000110112220233330
Define a soft set FA of X by Table 2.
Then FA is a soft subalgebra of X. For soft set FA, T={3}. Let GB be a soft set over X given by Table 3.
Then GB is a soft S-extension of X. But it is not a soft U1-translation of FA for any nonempty U1⊆T.
X
0
1
2
3
FA
{0,1,2}
{0,1}
{0,2}
{1,2}
X
0
1
2
3
GB
{0,1,2}
{0,1,2}
{0,2}
{0,1,2}
For a soft set FA of X, U1⊆T and U2∈P(X) with U2⊇U1, let
(9)UU1(FA;U2):={x∈X∣FA(x)⊇U2-U1}.
If FA is a soft subalgebra of X, then it is clear that UU1(FA;U2) is a subalgebra of X for all U2∈P(X) with U2⊇U1. But, if we do not give condition that FA is a soft subalgebra of X, then UU1(FA;U2) may not be a subalgebra of X as seen in the following example.
Example 15.
Let X={0,1,2,3,4} be a BCI/BCK-algebra presented as follows:
(10)*01234000000110000221000331100443310
Define a soft subset FA of X by Table 4.
Then FA is not a soft subalgebra of X with T={1}. Since FA(3*4)={0}⊉{0,4}=FA(3)∩FA(4) For U2={1,4} and U1={1}, we obtain UU1(FA;U2)={3,4} which is not a subalgebra of X since 3*3=0∉UU1(FA;U2).
X
0
1
2
3
4
FA
{0}
{0,2}
{0,2,3}
{0,3,4}
{0,4}
In the following theorem, relationship between U1-translations and UU1(FA;U2) is studied in case of soft subalgebra.
Theorem 16.
Let FA be a soft set over X and U1⊆T. Then the soft U1-translation FU1T of FA is a soft subalgebra of X if and only if UU1(FA;U2) is a subalgebra of X for all U2∈P(U) with U2⊇U1.
Proof.
Assume that the soft U1-translation FU1T of FA is a soft subalgebra of X. Then by Theorem 9, FA is a soft subalgebra of X if FU1T is a soft subalgebra of X. Further let a,b∈UU1(FA;U2), then FA(a)⊇U2-U1 and FA(b)⊇U2-U1 are subalgebras of X for all U2∈P(U) with U2⊇U1. Consider
(11)FA(a*b)⊇FA(a)∩FA(b)⊇U2-U1.
Therefore a*b∈UU1(FA;U2), which shows that UU1(FA;U2) is a subalgebra of X, for all U2⊆P(U), with U2⊇U1.
Conversely, suppose that UU1(FA;U2) is a subalgebra of X for all U2⊆P(U) with U2⊇U1. Now assume that there exist a,b∈X such that
(12)FU1T(a*b)⊂U2⊆FU1T(a)∩FU1T(b).
Then FA(a)⊇U2-U1 and FA(b)⊇U2-U1 but FA(a*b)⊂U2-U1. This shows that a,b∈UU1(FA;U2) and a*b∉UU1(FA;U2), which is a contradiction and so FU1T(a*b)⊇FU1T(a)∩FU1T(b) for all a,b∈X. Hence FU1T is a soft subalgebra of X.
Theorem 17.
Let FA be a soft subalgebra of X and let U1,U2⊆T. If U1⊇U2, then the soft U1-translation FU1T of FA is a soft S-extension of the soft U2-translation FU2T of FA.
Proof.
Since U1⊇U2, this implies FU1T(x)⊇FU2T(x), for all x∈X. So U1-translation is an extension of U2-translation, and from Theorem 9, FU1T and FU2T are soft subalgebras of FA. Hence soft U1-translation FU1T of FA is a soft S-extension of the soft U2-translation FU2T of FA.
For every soft subalgebra FA of X and U2⊆T, the soft U2-translation FU2T of FA is a soft subalgebra of X. If GB is a soft S-extension of FU2T and then there exists U1⊆T such that U1⊇U2 and GB(x)⊇FU1T(x), for all x∈X. Thus, we have the following theorem.
Theorem 18.
Let FA be a soft subalgebra of X and U2⊆T. For every soft S-extension GB of soft U2-translation FU2T of FA, there exists a U1⊆T such that U1⊇U2 and GB are a soft S-extension of U1-translation of FA.
Proof.
For every soft subalgebra FA of X and U2⊆T, the soft U2-translation FU2T of FA is a soft subalgebra of X. If GB is a soft S-extension of FU2T and then there exists U1⊆T such that U1⊇U2 and GB(x)⊇FU1T(x), for all x∈X. Then by Theorem 17, GB is a soft S-extension of U1-translation of FA.
Definition 19.
A soft S-extension GB of a soft subalgebra FA of X is said to be normalized if there exists x0∈X such that GB(x0)=X.
Definition 20.
Let FA be a soft subalgebra of X. A soft set GB of X is called a maximal soft S-extension of FA if it satisfies the following conditions:
GB is a soft S-extension of FA,
there does not exist another soft subalgebra of X which is a soft extension of GB.
Example 21 (see [<xref ref-type="bibr" rid="B34">33</xref>]).
Let Z+ be a set of positive integers and let “*” be a binary operation on Z+ defined by
(13)x*y=x(x,y),∀x,y∈Z+, where (x,y) is the greatest common divisor of x and y. Then (Z+;*,1) is a BCK-algebra. Let FA and GB be soft sets of Z+ which are defined by FA(x)={1,2,3} and GB(x)=Z+ for all x∈Z+. Clearly, FA and GB are soft subalgebras of Z+. By using definition of maximal soft S-extension, then it is easy to see that GB is a maximal soft S-extension of FA.
Proposition 22.
If a soft set GB of X is a normalized soft S-extension of a soft subalgebra FA of X, then GB(0)=X.
Proof.
Assume that GB is a normalized soft S-extension of a soft subalgebra FA of X then there exists x0∈X such that GB(x0)=X, for some x0∈X. Consider
(14)GB(0)=GB(x0*x0)⊇GB(x0)∩GB(x0)=X.
This implies GB(0)=X.
Theorem 23.
Let FA be a soft subalgebra of X. Then every maximal soft S-extension of FA is normalized.
Proof.
This follows from the definitions of the maximal and normalized soft S-extensions.
4. Soft Translations of Soft Ideals in Soft BCI/BCK-Algebras
Now concept of translation of a soft ideal of a BCI/BCK-algebra is introduced.
Definition 24.
A soft subset FA of a BCI/BCK-algebra is called a soft ideal of X, denoted by FA⊲SX, if it satisfies:
(∀x∈X)(FA(0)⊇FA(x)),
(∀x,y∈X)(F(x)⊇(FA(x*y)∩FA(y))).
Theorem 25.
If FA is a soft subset of X, then FA is a soft ideal of X if and only if soft U1-translation FU1T of FA is a soft ideal of X for all U1⊆T.
Proof.
Assume that FA⊲SX and let U1⊆T. Then FU1T(0)=FA(0)∪U1⊇FA(x)∪U1=FU1T(x) and
(15)FU1T(x)=FA(x)∪U1⊇(FA(x*y)∩FA(y))∪U1=(FA(x*y)∪U1)∩(FA(y)∪U1)=FU1T(x*y)∩FU1T(y)∀x,y∈X.HenceFU1T⊲SX.
Conversely, assume that FU1T is a soft ideal of X for some U1⊆T. Let x,y∈X. Then
(16)FU1T(0)⊇FU1T(x)⟹FA(0)∪U1⊇FA(x)∪U1⟹FA(0)⊇FA(x)byLemma6,
and so FA(0)⊇FA(x). Next
(17)FA(x)∪U1=FU1T(x)⊇FU1T(x*y)∩FU1T(y)=(FA(x*y)∪U1)∩(FA(y)∪U1)=(FA(x*y)∩FA(y))∪U1,
which implies that FA(x)⊇FA(x*y)∩FA(y) (by Lemma 6). Hence FA is a soft ideal of X.
5. Soft Extensions and Soft Ideal Extensions of Soft Subalgebras
In this section concept of soft ideal extension is being introduced and some of its properties are studied.
Definition 26.
Let FA and GB be the soft subsets of X. Then GB is called the soft ideal extension of FA, if the following conditions hold:
GB is a soft extension of FA.
FA⊲SX⇒GB⊲SX.
For a soft subset FA of X, U1⊆T and U2∈P(X) with U2⊇U1, define EU1(FA;U2):={x∈X∣FA(x)∪U1⊇U2}.
It is clear that if FA⊲SX, then UU1(FA;U2)⊲X for all U2∈P(U) with U2⊇U1.
Theorem 27.
For U1⊆T, let FU1T be the soft U1-translation of FA. Then the following are equivalent:
FU1T⊲SX.
(∀U2∈P(U))(U2⊃U1⇒EU1(FA;U2)⊲X).
Proof.
(1)⇒(2) Consider FU1T⊲SX and let U2∈P(U) be such that U2⊃U1. Since FU1T(0)⊇FU1T(x) for all x∈X, we have
(18)FA(0)∪U1=FU1T(0)⊇FU1T(x)=FA(x)∪U1⊇U2,
for x∈EU1(FA;U2).
(19)Hence0∈EU1(FA;U2).
Let x,y∈X be such that x*y∈EU1(FA;U2) and y∈EU1(FA;U2). Then FA(x*y)∪U1⊇U2 and FA(y)∪U1⊇U2, that is, FU1T(x*y)=FA(x*y)∪U1⊇U2 and FU1T(y)=FA(y)∪U1⊇U2. Since FU1T⊲SX, it follows that
(20)FA(x)∪U1=FU1T(x)⊇FU1T(x*y)∩FU1T(y)⊇U2,
that is, FA(x)∪U1⊇U2 so that x∈EU1(FA;U2). Therefore EU1(FA;U2)⊲X.
(2)⇒(1) Suppose that EU1(FA;U2)⊲X for every U2∈P(U) with U2⊇U1. If there exists x∈X with U3⊇U1 such that FU1T(0)⊂U3⊆FU1T(x) and then FA(x)∪U1⊇U3 but FA(0)∪U1⊂U3. This shows that x∈EU1(FA;U2) and 0∉EU1(FA;U2). This is a contradiction, and so FU1T(0)⊇FU1T(x), for all x∈X.
Now assume that there exist a,b∈X such that FU1T(a)⊂U4⊆FU1T(a*b)∩FU1T(b). Then FA(a*b)∪U1⊇U4 and FA(b)∪U1⊇U4, but FA(a)∪U1⊂U4. Hence a*b∈EU1(FA;U4) and b∈EU1(FA;U4), but a∉EU1(FA;U4). This is impossible and therefore FU1T(x)⊇FU1T(x*y)∩FU1T(y), for all x,y∈X. Consequently FU1T⊲SX.
Theorem 28.
Let FA⊲SX and U1,U2⊆T. If U1⊇U2, then the soft U1-translation FU1T of FA is a soft ideal extension of the soft U2-translation FU2T of FA.
Proof.
Since
(21)FU1T(x)=FA(x)∪U1,FU2T(x)=FA(x)∪U2,U1⊇U2, this implies that (FU1T(x)⊇FU2T(x))(∀x∈X). This shows that FU1T is a soft extension of FU2T.
Now, let FU2T is a soft ideal of X, then FU1T(0)=FA(0)∪U1⊇FA(x)∪U1=FU1T(x) for all x∈X, so we have (FU1T(0)⊇FU1T(x)). Consider
(22)FU1T(x)=FA(x)∪U1⊇(FA(x*y)∩FA(y))∪U1=(FA(x*y)∪U1)∩(FA(y)∪U1)=FU1T(x*y)∩FU1T(y)for all x,y∈X.
That is (FU1T(x)⊇FU1T(x*y)∩FU1T(y))(∀x,y∈X) so FU1T is a soft ideal of X. Hence FU1T is a soft ideal extension of FU2T.
6. Conclusion
Soft set theory is a mathematical tool to deal with uncertainties. Translation and extension are very useful concepts in mathematics to reduce the complexity of a problem. These concepts are frequently employed in geometry and algebra. In this papers, we presented some new notions such as soft translations and soft extensions for BCI/BCK-algebras. We also examined some relationships between soft translations and soft extensions. Moreover, soft ideal extensions and translations have been introduced and investigated as well. It is hoped that these results may be helpful in other soft structures as well.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
Authors are grateful to referees and Professor Feng Feng, the Lead Guest Editor of this special issue, for his kind suggestions to improve this paper.
MajiP. K.BiswasR.RoyA. R.Soft set theoryMolodtsovD.Soft set theory: first resultsAliM. I.FengF.LiuX.MinW. K.ShabirM.On some new operations in soft set theoryMajiP. K.RoyA. R.BiswasR.An application of soft sets in a decision making problemAktaşH.ÇağmanN.Soft sets and soft groupsChenD.TsangE. C. C.YeungD. S.WangX.The parameterization reduction of soft sets and its applicationsKongZ.GaoL.WangL.Comment on “a fuzzy soft set theoretic approach to decision making problems”FengF.LiC.DavvazB.AliM. I.Soft sets combined with fuzzy sets and rough sets: a tentative approachFengF.LiuX.Leoreanu-FoteaV.Soft sets and soft rough setsAliM. I.A note on soft sets, rough soft sets and fuzzy soft setsParkC. H.JunY. B.ÖztürkM.Soft WS-algebrasJunY. B.ParkC. H.Applications of soft sets in ideal theory of BCK/BCI-algebrasAygünoǧluA.AygünH.Introduction to fuzzy soft groupsÇağmanN.EnginoğluS.Soft matrix theory and its decision makingÇağmanN.EnginoS.Soft set theory and uni-int decision makingFengF.LiY.ÇağmanN.Generalized uni-int decision making schemes based on choice value soft setsFengF.LiY.Soft subsets and soft product operationsFengF.FujitaH.JunY. B.KhanM.Decomposition of fuzzy soft sets with finite value spacesLiuX.FengF.JunY. B.A note on generalized soft equal relationsLiuX.FengF.ZhangH.On some nonclassical algebraic properties of interval-valued fuzzy soft setsMaX.KimH. S.(M;N)-soft intersection BL-algebras and their congruencesMuhiuddinG.FengF.JunY. B.Subalgebras of BCK/BCI-algebras based on cubic soft setsSezerA. S.A new view to ring theory via soft union rings, ideals and bi-idealsSezerA. S.AtagünA. O.ÇağmanN.Soft intersection near-rings with its applicationsXinX.LiW.Soft congruence relations over ringsImaiY.IsekiK.On axiom systems of propositional calculi, XIVJunY. B.SongS. Z.Fuzzy set theory applied to implicative ideals in BCK-algebrasJunJ. B.XinX. L.Involutory and invertible fuzzy BCK-algebrasJunY. B.MengJ.Fuzzy commutative ideals in BCI-algebrasMengJ.JunY. B.KimH. S.Fuzzy implicative ideals of BCK-algebrasJunY. B.Soft BCI/BCK-algebrasMengJ.JunY. B.LeeK. J.JunY. B.DohM. I.Fuzzy translations and fuzzy multiplications of BCK/BCI-algebras