GJSFR Classification – F (FOR) 010101,010105,010206
When Is An Algebra Of Endomorphisms An
Incidence Algebra?
1Viji M., R.S.Chakravarti2
Abstract-Spiegel and O’Donnell give a characterization of
algebras of n×n matrices which are isomorphic to incidence
algebras of partially ordered sets with n elements. We
generalize this result to get a characterization of algebras of
endomorphisms of a vector space which are isomorphic to
incidence algebras of lower finite partially ordered sets.
AMS Subject Classification: 16S50.
Keywords-incidence algebra, partially ordered set, lower
finite, endomorphism.
I. INTRODUCTION
A partially ordered set X is said to be locally finite if, the
subset Xyz = {x X : is finite for each y, z
X such that is said to be lower finite if the subset
is finite for each z X and is said to
be upper finite if the subset is finite
for each z X. If a partially ordered set X is lower or upper
finite then it is clearly locally finite.
The Incidence algebra I(X,R) of a locally finite partially
ordered set X over the commutative ring R with identity is
with operations defined by
for all f, g I(X,R), r R and x, y, z X. The identity
element of I(X,R) is
The JacobsonRadical, denoted by J(T) of a ring with identity
is the intersection of all its maximal right ideals. This is
always a two sided ideal and it is the largest ideal J of the
ring T such that 1 − t is invertible for all t J. It is proved
([1], Theorem 4.2.5) that the Jacobson Radical of an
incidence algebra consists of all the functions f X. So we
have,
About1Viji M.,Dept. of Mathematics, St.Thomas’ College, Thrissur-680001,
Kerala E-mail:[email protected] 2R.S.Chakravarti, Dept. of Mathematics, Cochin University of Science and
Technology, Cochin-682022, Kerala.
E-mail:[email protected]
Proposition 1.- ([1], Cor.4.2.6) Let X be a locally finite
partially ordered set
and R a commutative ring with identity. Then
The following result gives a relation between multiplication
in an incidence algebra and matrix multiplication.
Proposition 2-([1], Proposition 1.2.4) Let X be a locally
finite partially ordered set and R a commutative ring with
identity. Then I(X,R) is isomorphic to a subring of M|X|(R),
the R−module of all maps from X×X to R with pointwise
addition and scalar multiplication.
Then a natural question that arises is that, which subalgebras
of M|X|(R) are incidence algebras? For incidence algebras of
finite posets over a field, we have the following
characterization,
Theorem 1-([1], Theorem4.2.10) Let K be a field and S a
subalgebra of Mn(K). Then there is a partially ordered set X
of order n such that I(X,K) if and only if
i. S contains n pairwise orthogonal idempotents, and
ii. is commutative.
II. A CHARACTERIZATION of I(X,K) WHERE X IS a
LOWER FINITE PARTIALLY ORDERED SET AND K IS a
FIELD
Theorem 2- Let V be a K−vector space with dimension |X|,
for a suitable set X. Let S be a subalgebra of EndKV . Then
there exists a lower finite partial ordering in X such that S
I(X,K) if and only if,
i. 1 S
ii. S/J(S) is commutative
iii. For each x X, there is an Ex S of rank 1, such
that
iv. Xy = is finite for each y X
Proof-First we prove that the conditions given are
sufficient. Let S be A subalgebra of EndKV satisfying
conditions (1), (2), (3) and (4). From condition (3) it is clear
that we may find a basis for V such that
Define Exy EndK(V ) by Exy(vz)
Define an order in X by if and only if Exy S
for all x X, is reflexive
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antisymmetric.
Now from condition (4), it is clear that is a lower
finite partially ordered set, since
We prove that Observe that, for each
we have by writing
So
It is clear that will preserve addition and scalar
multiplication and will map identity of S to identity in
I(X,K). Now we will prove that will preserve
multiplication.
for two elements P,Q S.
Then,
So that
So preserves multiplication, and hence is a
homomorphism. If then pxy = qxy for each pair
x, y X and this implies P = Q. So is injective. Let
Define T such that (this
is possible since X is lower finite). Clearly, (T) = f.
Hence is surjective and is an isomorphism from S to
I(X,K).
Now we have to prove that conditions (1), (2), (3) and (4)
are necessary. Suppose that I(X,K) S where S is a
subalgebra of EndK(V ) of a K−Vector
space V , with dimension |X|, where X is a lower finite
partially ordered set. Let us consider the map : I(X,K) !
EndKV where is the map that is defined in the proof of
sufficiency part.Thus where Tf is such that
will be isomorphic to
I(X,K) and will satisfy the conditions (1), (3), (4) clearly and
(2) follows from Proposition 1 and the fact that
Will also have the properties specified in conditions (1), (2),
(3) and (4). Hence the theorem.
III. REFERENCES
1) E.Spiegel and C.J.O‘Donnell ; Incidence Algebras,
Monographs and Textbooks in Pure and Applied
Mathematics, Vol.206, Marcel Dekker, New York,
1997.
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