Characterizations of Algebras
The so-called ideal and subalgebra and some additional concepts of algebras are discussed. A partial order and congruence relations on algebras are also proposed, and some properties are investigated.
In the past years, fuzzy algebras and their axiomatization have become important topics in theoretical research and in the applications of fuzzy logic. The implication connective plays a crucial role in fuzzy logic and reasoning [1, 2]. Recently, some authors studied fuzzy implications from different perspectives . Naturally, it is meaningful to investigate the common properties of some important fuzzy implications used in fuzzy logic. Consequentially, Professor Wu  introduced a class of fuzzy implication algebras, FI-algebras for short, in 1990.
In the past two decades, some authors focused on FI-algebras. Various interesting properties of FI-algebras [5, 6], regular FI-algebras [4, 7, 8], commutative FI-algebras , -FI-algebras , and other kinds of FI-algebras  were reported, and some concepts of filter, ideal, and fuzzy filter of FI-algebras were proposed [4, 12–18]. Relationships between FI-algebra and BCK-algebra [19, 20], MV-algebras , and Rough set algebras [22, 23] were partly investigated, and FI-algebras were axiomatized . In the recent work, the relationships between these FI-algebras and several famous fuzzy algebras were systematically discussed [25–30].
In 1900, Hilbert proposed the famous problem , which is the tenth problem of Hilbert. argues whether there is a useful approach to determine that the general Diophantine equation is solvable. Until 1970, this problem had been solved by Y. Matiyasevich. It is clear that there is not a method to judge whether the Diophantine equation can be solved. In 1987, Siekmann and Szabó studied the unification problem related to . It was concluded that unification problem of -rewriting systems cannot be predicated. And it was stated that, for every axiomatic system satisfying , it is impossible to determine the problem of – unification problem , where is a set of term combined by the symbols and .
Our work constructed a new algebra system. It was called the algebra, which is more general than the -rewriting systems. We studied the basic properties of algebra . In addition, if the operation is idempotent, then is a rewriting system. In this case, it is impossible to determine the problem of algebra unification problem. Analogously, if the operation is nilpotent, then is associated BCI-algebra.
The present paper is organized as follows: In Section 2, we review some basic concepts and the main properties of algebras and show relationship between the algebras and the related fuzzy logic algebras. Then Sections 3–6 contain main results of inverse semigroup of algebra, partial order on semigroup of algebra, congruence on semigroup of algebra, and ideal and subalgebra of algebra which will be considered, respectively.
2. Basis Concepts and Properties of Algebras
In this part, we firstly review some relevant concepts and definitions.
Definition 1 (see ). Let be a universe set, and , and let be a binary operation on . A -type algebra is called a fuzzy implication algebra, shortly, FI-algebra, if the following five conditions hold for all : (I1) ; (I2) ; (I3) ; (I4) if , then ; (I5) , where .Then algebra is an algebra of type . The notion was first formulated in 1996 by Deng and Xu and some properties were obtained (see ). This notion was originated from the motivation based on fuzzy implication algebra introduced by Wu (see ). He proved that, in a fuzzy implication algebra , the order relation satisfying if and only if is a partial order. In , Deng and Xu introduced a binary operation defined on fuzzy implication algebra , such that, for all ,where is an adjoint pair on .
In the corresponding fuzzy logic, the operation is recognized as logic connective “conjunction” and is considered as “implication.” If the above expression holds for a product , then is the residunm of . For a product the corresponding residunm is uniquely defined by
Let us note that is the greatest element of the set .
We proved that if , the following hold: then is a semigroup.
In fact, the multiplication defined as above is associative.
It was shown in  that, for a fuzzy implication algebra , considering any , there is :
Since , we have . So it can be concluded that is a semigroup.
By generalizing the expressions (3), we obtain the basic equations of algebra.
Definition 2 (see ). algebra is a nonempty set with a constant 0 and two binary operations , for all , satisfying the axioms: , , .
By substituting and in expressions and with and , respectively, we arrive at the expressions (3).
Theorem 3 (see ). Let be algebra. Then for all , the following holds:(1);(2), ;(3), .
Therefore the algebra is an algebra system with a pair of dual semigroups. Several interesting properties of algebra have been discussed earlier (see [32, 33]). Throughout this paper, we will denote the semigroup of algebra by .
Theorem 5. Let be algebra and let the following hold for every in : ; then is a commutative semigroup.
Proof. Since , this shows we have . This implies that . This yields is a commutative semigroup.
So in algebra , for any in , if holds, then and are the same and are a commutative monoid as well.
Definition 6 (see ). A residuated poset is a structure such that is a bounded poset, is a commutative monoid, it satisfies the adjointness property; that is,
Remark 7. Let be algebra. If is semigroup with the induced order , that is, , for all , where , and , for all , then semigroup is a residuated poset.
Remark 8. If a fuzzy implication algebra is with a partial order “”, such that any , , , and for all , the following conditions hold:then is algebra.
Remark 9. Let be algebra; then semigroups and are a pair of dual semigroup. A pair of dual operations form an adjoint pair ; that is, , for every .
Theorem 10. Let be algebra and for every in , then the following hold:
Proof. With Theorem 3, we have in . Hence, . Similarly, .
It is well known that partial ordering and congruences play an important role in the theory of semigroups. In this paper, we have given a partial order relation and a congruence on semigroup of algebra.
Definition 11 (see ). A semigroup is called(i)medial if it satisfies the equation ;(ii)trimedial (dimedial, resp.) if every subsemigroup of generated by at most three (two, resp.) elements is medial;(iii)left (right or middle, resp.) semimedial if it satisfies the identity , resp.);(iv)semimedial if it is both left and right semimedial;(v)strongly semimedial if it is semimedial and middle semimedial;(vi)exponential for every positive integer ;(vii)left (right, resp.) distributive if it satisfies .
Theorem 12. (1) Every semigroup of algebra is medial and semimedial.
(2) Every semigroup of algebra is strongly semimedial.
(3) Every medial semigroup of algebra is exponential.
Proof. It is easy.
3. On Inverse Semigroup of Algebra
We will assume the reader is familiar with elementary inverse semigroup theory as described in . Some properties of inverse semigroup have been discovered in literature (see [36, 37]). We present here some related definitions and examples of inverse semigroup of algebra, as follows.
Definition 13. A semigroup of algebra is called an inverse semigroup if for every there exists such that and . Henceforth, we define as the inverse of .
Theorem 14. The inverses of semigroup of algebra are not unique.
Proof. If is an inverse of and is an inverse of then and imply that
If and are both inverse of , then we can show thatIf , thenIn general, the expression does not hold. In this case, the inverse is not unique.
We will give some examples in the next section.
Example 15. Let be equipped with the operation defined by the following Caylay’s table:Then is algebra, but is not an inverse semigroup, because are inverse of or , and is inverse of , and is inverse of , but has no inverse.
Example 16. Let be equipped with the operation defined by the following Caylay’s table:It is clear that the above system forms an inverse semigroup. Obviously, are inverse elements of or or , and is inverse of , and is inverse of in
Clearly is algebra, and is an inverse semigroup.
4. Partial Order on Semigroup of Algebra
We will now investigate the partial order on semigroup of algebra.
If , are elements of a semigroup in algebra , then by we mean , where is an idempotent in .
If and are idempotents in , we haveby repeatedly using . That means the product of idempotents is an idempotent.
Theorem 17. The relation is partial order relation on an inverse semigroup of algebra .
Proof. Since is an inverse semigroup of algebra , for each , there exists such that . Now . This implies that is an idempotent. Hence for . So .
Let . This implies that , where is an idempotent. Let ; then . If is an idempotent, we have and . This shows that is antisymmetric.
Let and ; then there exist idempotent and such that and . So . This implies that as is an idempotent.
Theorem 18. If is an inverse semigroup of algebra , , then(1) implies and , for all ,(2) implies ,(3) and ; then .
Proof. (1) If then for an idempotent , . Now implies that . Hence . Also, since is an idempotent, . Hence .
(2) Now implies that, for an idempotent , . So . Since is an idempotent, hence .
(3) Let and . Then we have idempotents and such that and . Now implies that . This completes the proof.
Theorem 19. Let and be two associative bands of algebra, and . Then implies that and implies that .
Proof. Suppose that ; then .
And in an associative band , suppose that ; then .
5. Congruence on Semigroup of Algebra
Analogous characterization of the maximum idempotent-separating congruence on an eventually orthodox semigroup is given (see ). As important consequences, some sufficient conditions for an eventually regular subsemigroup of to satisfy are obtained, whence if is fundamental, then so is . Similarly to the treatises by Mitsch  and Luo and Li (see ), we consider a semigroup of algebra. Let . If implies that , then the relation is called left compatible. If implies that , then the relation is called right compatible. If and imply that then the relation is called compatible. A compatible equivalence relation is called a congruence. In this paper, we have defined a congruence relation on an inverse semigroup of algebra.
If is a congruence relation on an inverse semigroup of algebra, then we can define a binary operation in in a natural way as .
If and for , then we have and and hence . Thus the binary operation is well-defined. consists of elements of the form , where .
Let and be in ; then is in . Now if then . If , then . This shows that and are inverse of each other. So is an inverse semigroup on condition that is an inverse semigroup.
Let be an idempotent in an inverse semigroup ; then belongs to and implies that is idempotent in .
Theorem 20. Let be a congruence relation on an inverse semigroup of algebra. If is an idempotent in , then there exists an idempotent such that
Proof. If is an idempotent in then . This implies that and . Let be an inverse of in . Then and .
If , then . So is an idempotent. Now implies . Again and imply that or or . So and imply that
6. Ideal and Subalgebra on Semigroup of Algebra
Recently, Jun and Song (see ) considered int-soft (generalized) bi-ideals of semigroups, further properties and characterizations of int-soft left (right) ideals are studied, and the notion of int-soft (generalized) bi-ideals is introduced. Relations between int-soft generalized bi-ideals and int-soft semigroups are discussed, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are considered. Refer to the literature (see [37, 40–42]); in this paper, we focus our attention on the ideal and subalgebra on semigroup of algebra.
The semigroup of algebra determines an order relation on : , where is an element that does not depend on the choice of .
Definition 21. A nonempty subset of algebra is called an ideal of if it satisfies (1) , (2) () () ().
Lemma 22. An ideal of algebra has the following property: .
Proof. If , , , then ; by Definition 21, we get .
Definition 23. Let be algebra. For any , let . Then is said to be complicated if for any , the set has a greatest element.
Theorem 24. Let be algebra and let be a subset of . Then is an ideal of if and only if for all .
Proof. () is an ideal and are elements of . For any , we have . Hence we have with Lemma 22. Then we obtain since is an ideal.
() If , for all , we have since . For any and , let . Then . Hence, since , we obtain . Hence, .
Theorem 25. Let be algebra, for any , defining , the following statements are true:(1)If , then .(2)If , then .(3)If , then .(4)If , and , then .
Proof. (1) If , then , , .
(2) If , then , so ; hence .
(3) Assume that implies . Thus .
(4) If , then ; we have , ; then , and we have ; we get . Hence .
Definition 26. Let be algebra and a nonempty subset of . Then is called a subalgebra of , if and whenever .
Theorem 27. Let be algebra, for any , if ; then is a subalgebra of .
Proof. Let , then ; that is, and imply , so ; that is, , can be proved in a similar way. Therefore, if , then is a subalgebra of .
Theorem 28. Let be algebra, is a subalgebra of , if the condition , is satisfied, and then any ideal of is algebra.
Proof. For any , we have , if is ideal of ; we obtain with ; we get . Hence , . So is algebra, where is an ideal of .
Theorem 29. Let be algebra and . Then the set is ideal of and is a subalgebra of as well.
Proof. (1) For any , we have , so . , , there is , and ; we get , so we have . Hence, is ideal of .
(2) If , then , then , and . Similarly it can be proved that . Hence, is a subalgebra of .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by NSF of China (11305097).
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