Research Article | Open Access
M. Yazi, "Ordered Structures and Projections", International Journal of Mathematics and Mathematical Sciences, vol. 2008, Article ID 783041, 6 pages, 2008. https://doi.org/10.1155/2008/783041
Ordered Structures and Projections
We associate a covering relation to the usual order relation defined in the set of all idempotent endomorphisms (projections) of a finite-dimensional vector space. A characterization is given of it. This characterization makes this order an order verifying the Jordan-Dedekind chain condition. We give also a property for certain finite families of this order. More precisely, the family of parts intervening in the linear representation of diagonalizable endomorphism, that is, the orthogonal families forming a decomposition of the identity endomorphism.
In this paper, we consider a poset , there is a set of all idempotent endomorphisms (projections) of a finite-dimensional vector space endowed with a reflexive, symmetric, and transitive binary relation (denoted ) ([1, 2]). Given two elements and of , is equivalent to and We say that covers (denoted ) if , and the interval is empty. So an element of is an atom (resp., a coatom) if coverers (resp., covered by ) where zero endomorphism (resp., identity endomorphism) is the least element (resp., the greatest element) of . When the bounds of two elements and exist we denote their meet (resp., their join) by (resp., ) ([3, 4]). We remark that the usual order relation defined in between two elements and is expressed by relations of inclusion between their kernels and their ranges, then between elements of : set of subspaces of which is well known that when this later is endowed with the relation of set inclusion, the couple is a geometric lattice ([5–7]) where the covering relation is defined for two subspaces and by and
The extension of the notion of covering constitutes my principal motivation in this paper. Indeed, the main result of this paper, given in Section 2, is a characterization of covering relation defined by means of the one from ([6, 8]). We use this result to show that is a graded poset with the rank function defined by for all and all maximal chains between the same endpoints have the same finite length ([3, 4]). As a final point, when two elements and of satisfy we observe, as we will show in Section 3, that the poset possesses some properties, among other things is the covering property ([5, 6]).
2. Covering Relation
Proposition 2.1. If is of finite dimension, the ordered set verifies the ascending chain condition.Proof. Let be an increasing sequence of elements from . Then is equivalent to and Since is of finite dimension, then verifies the ascending chain condition, that is, there exists such that for all Hence, for all the elements are two by two isomorph. Since they are two by two comparable, they are equal. Hence, for all
We will prove later that this proposition is a consequence of Proposition 4.1.
Before stating a theorem, we remark that for two
elements and of such that one has this following equivalence: From this, we have the following theorem.
Theorem 2.2. If and are two elements of such that the two following properties are equivalent:(i);(ii)Proof. (i)(ii) Assume that does not cover that is, there exists in an element such that or equivalently, and Hence, we have a contradiction.
Conversely, assume that does not cover it follows from the precedent remark that does not cover
The chains of of respective endpoints and are then at least of length two. There exists then at least a couple of elements such thatwith and
Amongts these couples, we prove that it exists one which is the direct sum of that is From the relation it follows Then
Set with and a basis of a basis of and a basis of If we consider the union of the precedent bases, we obtain obviously a base of Thus, the subspaces and defined as follows: subspace generated by subspace generated by form, in fact, a couple of for which is a direct sum, that is, moreover, verifying the relations and Thus, the existence of the projection on along . Hence, we have a contradiction.Corollary 2.3. The elements of for which the image is a line are atoms. Dually, the elements of for which the image is a hyperplane are coatoms.
3. Covering Property
Several ordered sets having the property of lattice
satisfy the covering property (e.g., ). In the following result, we prove that one
also finds it in when two arbitrary elements commute.Proposition 3.1. Let and be two elements of . If is an atom such that then the covering property is verified, that
Dually, if is a coatom, then Proof. It is well known that and exist, and we have the following: If is an atom, the subspaces (resp., ) represent in an atom (line) (resp., a coatom is a hyperplane in ). So, the assumption impliesHence, in the equality (3.2) implies or equivalently Furthermore, the equality (3.3) implies or equivalently Then from Theorem 2.2
Using a similar reasoning as above we obtain Proposition 3.2. Let be three elements of such that If and , Dually, if and , Proof. From Theorem 2.2, and imply and , So, in we have and We have then In a similar reasoning we can prove that if and
4. Rank Function
Theorem 2.2 allows us to define in a rank function defined by We can easily prove thatProposition 4.1. is of finite length and of length equal to the dimension of Proof. Let a maximal chain of : By applying to this sequence, we get which proves that the length of is equal to the dimension of
being of finite length, it verifies the ascending and descending chain condition.Corollary 4.2. The poset with the rank function is of finite length, it verifies then the Jordan-chain condition, that is, all maximal chain between the same endpoints have the same finite length.Corollary 4.3. If and are two elements of such that Proof. This follows from a property of the dimension of an endomorphism of a finite-dimensional vector space.
It is well known that when , and are three elements commuting two by two, each one of them commutes with the supremum and infimum of the two others. We propose to generalize this result in the theorem below.Theorem 4.4. If is a family of elements of commuting two by two, that is, , thenIn particular, if this family is orthogonal, then Proof. For proving the equality we use induction on Since by assumption the elements commuting two by two, we have for . For Assume that the equality is verified until the order ,ThenIt follows from the combinatorial relation for all that Hence we have the desired equality.
It is clear that if the family is orthogonal (), the equality proved before becomes It is well known in Linear algebra that if is an orthogonal family verifying in addition , then this family characterizes diagonalizable endomorphisms.
The consequence of the preceding theorem can constitute a characterization of a diagonalizable endomorphism of a finite dimensional vector space by means of the ordered set of the idempotent endomorphism in the same space. It is thus natural to ask the following question. Can one in the general case for a given order of finite length satisfying in addition the condition of Jordan-Dedekind identify a finite family such that and for where and are the bounds of . If the answer is affirmative, can one in this case establish a junction between the Linear algebra and the ordered structures so that the identification of a diagonalizable endomorphism is concrete.
- L. Chambadal, Exercices et Problèmes d'Algèbre, Dunod, Paris, France, 2nd edition, 1978.
- P. R. Halmos, Finite-Dimensional Vector Spaces, The University Series in Undergraduate Mathematics, D. Van Nostrand, Princeton, NJ, USA, 1958.
- B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, UK, 1990.
- B. Leclerc, “Un cours sur les ensembles ordonnés,” Les Cahiers du CAMS 134, Centre d'Analyse et de Mathématique Sociales, Paris, France, 1997.
- G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, RI, USA, 3rd edition, 1967.
- F. Maeda and S. Maeda, Theory of Symmetric Lattices, Die Grundlehren der Mathematischen Wissenschaften, Band 173, Springer, New York, NY, USA, 1970.
- G. Szász, Introduction to Lattice Theory, Academic Press, New York, NY, USA, 3rd edition, 1963.
- B. Barbut and B. Monjardet, Ordre et Classification. Algèbre et Combinatoire, vol. I-II, Hachette Université, Paris, France, 1970.
Copyright © 2008 M. Yazi. 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.