Algebra

Volume 2014 (2014), Article ID 263605, 10 pages

http://dx.doi.org/10.1155/2014/263605

Research Article

Simplices in the Endomorphism Semiring of a Finite Chain

Ivan Trendafilov

Department of Algebra and Geometry, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, Bulgaria

Received 30 August 2013; Accepted 7 March 2014; Published 27 April 2014

Academic Editor: Andrei V. Kelarev

Copyright © 2014 Ivan Trendafilov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish new results concerning endomorphisms of a finite chain, if the cardinality of the image of such endomorphism is no more than some fixed number. The semiring of all such endomorphisms can be seen as a simplex whose vertices are the constant endomorphisms. We explore the properties of these simplices.

1. Introduction and Preliminaries

It is well known that each simplicial complex has a geometric (continuous) interpretation as a convex set spanned by geometrically independent points in some Euclidean space.

Here, we present an algebraic (discrete) interpretation of simplicial complex as a subsemiring, containing (in some sense spanned by) constant endomorphisms of the endomorphism semiring of a finite chain. The endomorphism semiring of a finite semilattice is well studied in [1–10].

The paper is organized as follows. After the introduction and preliminaries, in Section 2, we give basic definitions and obtain some elementary properties of simplices. Although we do not speak about any distance here, we define discrete neighborhoods with respect to any vertex of the simplex. In Section 3, we study discrete neighborhoods, left ideals, and right ideals of a simplex. The main results are Theorems 9 and 12, where we find two right ideals of simplex. In Section 4, Theorem 15 is the main result of the paper, where we show that important objects (idempotents, -nilpotent elements, left ideals, and right ideals) of simplex (big semiring) can be constructed using similar objects of coordinate simplex (little semiring).

Since the terminology for semirings is not completely standardized, we say what our conventions are. An algebra , with two binary operations + and on , is called a semiring if:

(i) is a commutative semigroup;(ii) is a semigroup;(iii)both distributive laws hold and , for any .

Let be a semiring. If a neutral element of the semigroup exists and or , it is called a left or a right zero, respectively, for all . If , for all , then it is called zero. An element of a semigroup is called a left (right) identity provided that or , respectively, for all . If a neutral element of the semigroup exists, it is called identity.

A nonempty subset of is called an ideal if , , and .

The facts concerning semirings can be found in [1].

For a join-semilattice , set of the endomorphisms of to be a semiring with respect to the addition and multiplication defined as follows:

(i), when , for all ;(ii), when , for all .

This semiring is called the endomorphism semiring of .

In this paper, all semilattices are finite chains. Following [2], we fix a finite chain and denote the endomorphism semiring of this chain with . We do not assume that for arbitrary . So, there is not a zero in endomorphism semiring . Subsemirings , where , of the semiring , consisting of all endomorphisms with fixed point , are considered in [3].

If such that , for any , we denote as an ordered -tuple . Note that the mappings will be composed accordingly, although we shall usually give preference to writing mappings on the right, so that means “first , then ”. The identity and all constant endomorphisms are obviously (multiplicatively) idempotents.

Let . For every endomorphism , the elements of

are called -nilpotent endomorphisms. An important result for -nilpotent endomorphisms is as follows.

Theorem 1 (see [4, Theorem 3.3]). For any natural , , and , the set of -nilpotent endomorphisms is a subsemiring of . The order of this semiring is , where is the th Catalan number.

Another useful result is as follows.

Theorem 2 (see [5, Theorem 9]). The subset of , , of all idempotent endomorphisms with fixed points , is a semiring of order .

For definitions and results concerning simplices, we refer the reader to [6, 7].

2. The Simplex

Let us fix elements , where , , and let . We consider endomorphisms such that . We denote this set by .

Let and consider the set

For , let , if and only if the sets and have a common least element. In this way, we define an equivalence relation. Any equivalence class can be identified with its least element which is the constant endomorphism , where .

Now take a simplicial complex with vertex set . The subset is a face of . Hence, we can consider the set as a face of . In particular, when the simplicial complex consists of all subsets of , it is called a simplex (see [6]) and .

It is easy to see that and imply and , and so we have proved the following.

Proposition 3. For any set , the simplex is a subsemiring of .

The number is called a dimension of simplex . Any simplex , where , is a face of simplex . If , face is called a proper face.

The proper faces of simplex are as follows:

(i)-simplices are vertices ;(ii)-simplices are called strings; they are denoted by , where ;(iii)-simplices are called triangles; they are denoted by , where ;(iv)-simplices are called tetrahedra; they are denoted by , where ;(v)the last proper faces are simplices , where .

The boundary of the simplex is a union of all its proper faces and is denoted by . The set

is called an interior of simplex .

It follows that the interior of simplex consists of endomorphisms , such that . So, we have the following.

Proposition 4. The interior is an additive semigroup.

Proposition 5. Any face of simplex is a left ideal.

Proof. Let be a face of simplex . Obviously the face is a subsemiring of . Let and . Since for any we have , then . Thus . Hence, is a left ideal of simplex .

Note that any face of some simplex is not a right ideal of the simplex. For instance, take the vertex . Then, , for all . From the last proposition consider the following.

Corollary 6. The boundary is a multiplicative semigroup.

The boundary and the interior of a simplex are, in general, not semirings.

For any natural , endomorphism semiring is a simplex with vertices . The interior of this simplex consists of endomorphisms , such that . Since the latter is valid only for identity , it follows that .

There is a following partial ordering of the faces of dimension of simplex : least face does not contain the vertex and the biggest face does not contain the vertex .

The biggest face of the simplex is the simplex . Now which is a subsemiring of . Similarly, the least face of is . Then, which is also a subsemiring of . The other faces of , where , do not have this property. Indeed, one middle face is . But set is not a semiring because, for any and any , if , then .

Let us fix vertex , where of simplex . The set of all endomorphisms such that just for elements is called th layer of the simplex with respect to , where . We denote the th layer of the simplex with respect to by . So, the -layer with respect to any vertex of the simplex is a face of the simplex; hence, it is a semiring. In the general case, the th layer , where , , is not a subsemiring of simplex .

On the other hand, since consists of all endomorphisms , such that just for elements , it follows that this th layer is closed under the addition. Hence, we have the following.

Proposition 7. Any layer of simplex is an additive semigroup.

Let be an arbitrary vertex of simplex . From a topological point of view, the set is a discrete neighborhood consisting of the “nearest points to point .” Similarly, we define . More generally

where and is called discrete -neighborhood of the vertex .

3. Subsemirings and Ideals of the Simplex

Lemma 8. Let , where , be a vertex of the simplex and be the th layer of the -simplex with respect to . Then, the set , where , is a subsemiring of .

Proof. We consider three cases.

Case 1. Let . Then, elements of are endomorphisms:

Since , it follows that set is closed under the addition.

We find , for all . Also we have , for all , with the only exception when . Now , , and . Hence, is a semiring.

Case 2. Let . Then, elements of are endomorphisms:

Since , it follows that the set is closed under the addition.

We find , for all . Also we have , for all , with also the only exception when . We have , , and . Hence, is a semiring.

Case 3. Let . Then elements of are endomorphisms:

Since , it follows that set is closed under the addition.

Now there are four possibilities.

Let and . Then

Since , for , and, in a similar way, , for , and also , it follows that is a commutative semiring.

Let and . Then, ,

We also observe that . All the other equalities between the products of the elements of are the same as in (1).

Let and . Then, ,

We also observe that . All the other equalities between the products of the elements of are the same as in (1).

Let and . Now all equalities between the products of the elements of are the same as in (1), (2), and (3). So, is a semiring.

Theorem 9. The union of the discrete 1-neighborhoods with respect to all vertices of the simplex is a right ideal of the simplex.

Proof. Let . Then, , , or , where and . Let . Then , , or , where and .

Suppose that . Then, we find or , where . So, in all cases , what means that is closed under the addition.

Let and . Then, , for some .

If and , then it follows that .

If , where , , and , then .

If , where , , and , it follows that .

Hence, in all cases .

Any simplex which is a face of simplex is called internal of the simplex , if and .

Similarly simplex , which is a face of -simplex , is called internal simplex, if and .

Immediately from the proof of Proposition 4 consider the following.

Corollary 10. For any internal simplex , semirings are commutative and all their elements are -nilpotent, where .

Lemma 11. Let , where , be a vertex of internal simplex . Then, the set , where , is a subsemiring of .

Proof. Since , it follows that the elements of are endomorphisms:

From Lemma 8, we know that the discrete 1-neighborhood is closed under the addition. From Proposition 7, it follows that the layer also is closed under the addition. Hence, in order to prove that is closed under the addition, we calculate the following:

where , , , and . So, we prove that the discrete 2-neighborhood is closed under the addition.

Now we consider six cases, where, for the indices, the upper restrictions are fulfilled.

Case 1. Let . We shall show that all endomorphisms of are -nilpotent with the only exception when . When , since is the least image of any endomorphism, there are only a few equalities: ,

Hence, it follows that is a commutative semiring with trivial multiplication.

If , it is easy to see that endomorphism is the unique idempotent of (see [5]). Now, we find , , and . Hence, is a semiring.

Case 2. Let . We shall show that all the endomorphisms of are -nilpotent with the only exception when . When , we find

If , the only idempotent is and we find

Hence, is a semiring.

Case 3. Let and . We find the following trivial equalities, which are grouped by duality:

Case 4. Let and . Then, is the only idempotent in . Additionally, to the equalities of the previous case, we find

Case 5. Let and . Now the only idempotent endomorphism in is . We additionally find the following equalities:

Case 6. Let and . Now, in , there are two idempotents: and . Here, the equalities from Cases 4 and 5 are valid and also all the equalities from Case 3, under the respective restrictions for the indices, are fulfilled.

Hence, is a semiring.

From Lemma 11, we find the following.

Theorem 12. The union of the discrete 2-neighborhoods with respect to all vertices of internal simplex is a right ideal of the simplex.

Proof. For the endomorphisms of and , where , from (11), it follows as shown in Table 1

Let , , be elements of (see Table 1). We denote by an endomorphism, which maps the same elements to , but other images (one or two) are not in . For example, , where and . Evidently, , for .

Now we calculate the following:

(1), or , , and or , , and or ;(2), or , , and or , , and or ;(3) or , or , or , or , or , and or ;(4), or , , and or , and or ;(5) or or , or or or , or or , or or or , or and or or ;(6) or , or , or or , or or , or and or or .

From these calculations and Lemma 11 we conclude that is closed under the addition.

Let and . Then, , for some . Considering the cases where , , we consider in the proof of Theorem 9. Now we consider three new cases.

Case 1. Let , where . Let , , and . Then, . The same is true when or .

Case 2. Let , where . Let , , and . Then . The same is true when or .

Case 3. Let , where and . Let , , and . Then, . The same is true when , , or .

Hence, in all cases, .

Table 1

From Theorem 9 and Theorem 12, we obtain the following.

Corollary 13. (a) If is an internal simplex, then the union of the discrete 1-neighborhoods with respect to all vertices is an ideal of the simplex.

(b) Let be an internal simplex and is internal of the simplex . Then, the union of the discrete 2-neighborhoods is an ideal of the simplex .

Proof. (a) If is an internal simplex and , then and . We calculate .

From Theorem 9, it follows that is an ideal of the simplex .

(b) If satisfies the condition of Theorem 12 and , then and . Thus (see the notations in the proof of Theorem 12) we obtain

From Theorem 12, it follows that is an ideal of the simplex .

Proposition 14. Let be a simplex.

(a)For the least vertex , it follows that .(b)For the biggest vertex , it follows that .

Proof. (a) Since is the least vertex of the simplex, it follows that layer consists of endomorphisms , where ; that is, . All the layers , where , consist of endomorphisms having as a fixed point. So, .

Conversely, let . Then, . Since is the least vertex of the simplex, we have ; that is, , where . Hence, .

(b) Since is the biggest vertex of the simplex, it follows that layer consists of endomorphisms , where . So, implies that the images of ,, are not equal to , but . For all the endomorphisms of layers , where , we have . Hence, the elements of these layers have as a fixed point and .

Conversely, let . Then, . Since is the biggest vertex of the simplex, we have . Thus, it follows that , where .

Hence, .

4. A Partition of a Simplex

Let be a simplex. Then, is called endomorphism of type , where , for , , if .

Obviously, the relation , if and only if and are of the same type, is an equivalence relation. Any equivalence class is closed under the addition. But there are equivalence classes which are not closed under the multiplication. For example, consider , where . Since , , , and , the type of is . But is of type .

Sometimes it is possible to describe the semiring structure of union of many equivalence classes, that is, blocks of our partition. For example, the union of endomorphisms from all the blocks of type is the set of such that and, from Proposition 14, it is the semiring .

The type of any endomorphism is itself an endomorphism of a simplex . The simplex is called a coordinate simplex of . From this point of view, the set of endomorphisms from all the blocks of type really corresponds to the set of all endomorphisms , which is the semiring . More generally, we can consider the semiring , where , consisting of all such that . Then, . So the union of endomorphisms from all the blocks of type is semiring . Now, more generally again, we can consider the semiring consisting of all endomorphisms of having as fixed points. Then, similarly, , for all . So the union of endomorphisms from all the blocks of type is semiring . The next results from this section are announced in [8].

Theorem 15. Let be a simplex. Let be a subsemiring of the coordinate simplex of . The set of endomorphisms of , of type , where the endomorphism , is a semiring. Moreover, when is a (right, left) ideal of semiring , it follows that is a (right, left) ideal of simplex .

Proof. Let . Let be of type , where the endomorphism and similarly are of type , where the endomorphism . Then, we find , where . But the endomorphism is the sum . So we prove that endomorphism is of type ; that is, .

Now, let us assume that and are arbitrary endomorphisms of . Then, we find

So, if is a right ideal of and , is a left ideal of and , or is an ideal of and one of and is from , it follows that . Hence, in each of the three cases and this completes the proof.

Now, let endomorphism be of type and the endomorphism from the coordinate simplex be an idempotent, different from constant endomorphisms , where , and the identity . Then we say that is of an idempotent type.

Corollary 16. The set of endomorphisms of a fixed idempotent type is a semiring.

Proof. Obviously, since the set , where is an idempotent, is a semiring, the semirings of an idempotent type are denoted by , where is the corresponding idempotent.

Let . For any , we consider (see the first section) the set

From Theorem 1, it follows that , for and , is a subsemiring of .

Now, let the endomorphism be of any type , where the endomorphism , for some fixed . Then, we say that is of an -nilpotent type.

Corollary 17. The set of endomorphisms of -nilpotent type, for some fixed , is a semiring.

Proof. Immediately from the last theorem and Theorem 1, the semirings of -nilpotent type are denoted by .

Now, let the endomorphism be of any type , where the endomorphism from the coordinate simplex is neither an idempotent nor an -nilpotent for some . Then, according to [5], is a root of some idempotent . Since the roots of identity of semiring do not exist (see [2]), it follows that is an idempotent, different from , , and identity. In this case, the endomorphism is called a type related to idempotent type . To clarify the above definition, we give an example of endomorphisms of type related to some idempotent type.

Example 18. Let us consider the simplex . The coordinate simplex consisting of all the types of endomorphisms of is the simplex . In this coordinate simplex, we chose the idempotent . From Theorem 19 of [5], it follows that the idempotent and its roots form a semiring of order . Let us note that Catalan sequence is the sequence , where . So, the elements of the semiring, generated by contains 5 endomorphisms: , , , , and . Now, we choose endomorphisms from the simplex : and . The endomorphism is of type , related to idempotent type , and endomorphism is of type , related to the same idempotent type. We compute which is an idempotent and also which is not an idempotent, but is an idempotent. Thus, we show that and are roots of different idempotents, but they are of types related to the same idempotent type. We can also show that is an idempotent and also is an idempotent.

Corollary 19. The set of endomorphisms of a type, related to some fixed idempotent type, is a semiring.

Proof. Immediately from the last theorem and Theorem 19 of [5], the semirings from the last corollary are called idempotent closures of type , where is the corresponding idempotent and is denoted by .

From the last theorem, we also find the following.

Corollary 20. The set of endomorphisms of , of type , where the endomorphism belongs to some face of the coordinate simplex , is a left ideal.

Now we describe the left ideals from the last corollary when .

Example 21. For a simplex , the coordinate simplex is and its faces are as follows:

 triangles: , , , and ; strings: , , , , , and ; vertices: , , , and .

Then, the left ideal of simplex consists of endomorphisms , such that is not a fixed point of . Now, the triangle , which is a left ideal of semiring , is contained in . Moreover,

Similarly, the left ideals ,  И  consist of endomorphisms , such that , , and are not fixed points of , respectively.

The left ideals , where , , consists of endomorphisms , such that and are not fixed points of . Let  И  . Then, . Observe that in the interior of one of the triangles with vertices , , and and , , and , respectively, there are endomorphisms , such that and are not fixed points of .

The left ideal , where is actually the ideal .

The left ideal consists of endomorphisms such that , , and are not fixed points of . Hence, all elements of this left ideal have as a fixed point. Similarly, we determine the left ideals , , and .

From the last theorem, also we have two consequences.

Corollary 22. The set of endomorphisms of the simplex , having a type , where , is an ideal.

Corollary 23. The set of endomorphisms of the simplex , having a type , where the endomorphism belongs to the union of the discrete 1-neighborhoods of all vertices of coordinate simplex , is a right ideal.

At last, we consider the endomorphisms of type . Now the corresponding endomorphism from the coordinate simplex is identity . In order to find the set of endomorphisms of this type, we need the following definition. Idempotent is called a boundary idempotent of the simplex, if , similarly, an interior idempotent of the simplex, if . Note that, in the coordinate simplex, the identity is the unique interior idempotent.

Theorem 24. The set of endomorphisms of , which are right identities, is a semiring of order .

Proof. Let us denote by semiring . Then, for any element and arbitrary endomorphism , it follows that , for some . Then ; that is, is an idempotent. Since , where , for any (if we suppose , then ), it follows that is an interior idempotent.

Conversely, let be an interior idempotent. Then, we can express , where , for any (if we suppose , then ). So, we prove that the semiring consists of all interior idempotents of the simplex.

Let and be an arbitrary element of the simplex. For , it follows that , where . Then . Hence, is a right identity of the simplex.

Conversely, let be a right identity of the simplex. Evidently, is an idempotent. Assume that, for some , we have . Since for some and , it follows that , where , we find ; that is, , which contradicts that is a right identity. So, ; that is, is an interior idempotent or .

Hence, we prove that is a semiring of right identities of simplex . Since elements of are all the idempotents with fixed points , from Theorem 2 (Section 1), it follows that .

So, we can construct a partition of the simplex such that all blocks of this partition are as follows:

(1)semirings , where ;(2)semirings , where is an idempotent of ;(3)semirings , where is an idempotent of ;(4)semiring .

From the last theorem we also find the following.

Corollary 25. There is at least one right identity of the simplex .

If there are two or more right identities of simplex , then there is not a left identity. Let us suppose that there is a left identity . Let be a right identity of the simplex. Then, it follows that . If there is a single right identity, from the last theorem, it follows that , for any . Then, this single right identity is . Let . We find , and so we have proved.

Corollary 26. There are not any left identities of the simplex .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is very much thankful to the Academic Editor for suggestions and support.

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