Affine Semigroups

Cohen, Haskell; Collins, H. S.

doi:10.2307/1993425

Cited by 7 publications

(12 citation statements)

References 6 publications

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“…The next result is a kissing cousin of results of Kakutani, Klee and Peck (see the discussion in Chapter V of [2]) and extends a result in [1]. THEOREM Particularizing this it follows that if T contains the set of extremal points then 5 has a left zero, thus K consists entirely of left zeroes and is convex.…”

Section: If S Is Compact Affine and If P 9^ • Then P Is A Closed Extrsupporting

confidence: 57%

Remarks on affine semigroups

Wallace¹

1960

Bull. Amer. Math. Soc.

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A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication, denoted by juxtaposition. In what follows S will denote one such and it will be assumed that S is compact. It thus entails no loss of generality to suppose that S is contained in a locally convex linear topological space 9C, but no particular imbedding is assumed. For general notions about semigroups we refer to [3] and for information concerning linear spaces to [2].It has been known for some time [3] that if 9C is finite dimensional, if S is convex (recall that S is compact) and if S has a unit (always denoted by u) then the maximal subgroup, H u , which contains u is a subset of the boundary of S relative to 9C.Let F denote the boundary of 5, K the minimal ideal of S and, for any subset A of 5, let P(A) = {x I x G S and xA = A}.As is customary, AB denotes the set of all products ab with aÇ^A and &G-Z? and we generally write x in place of \x\. It will be convenient to abbreviate P(S) by P. The structure of P is known in the following sense-supposing that PT^D the set PP\E^D and is indeed the set of left units of 5, E being the set of idempotents. Moreover, if e^.PC\E then Pe is a maximal subgroup of S and the assignment (#, y)->xy is an iseomorphism (topological isomorphism) of PeX(Pr\E) onto P. The following is a corollary to the principal result of [4]: THEOREM 1. If S is compact and convex and if ST^K then P(F) = P(S) C F.

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Section: If S Is Compact Affine and If P 9^ • Then P Is A Closed Extrsupporting

confidence: 57%

Remarks on affine semigroups

Wallace¹

1960

Bull. Amer. Math. Soc.

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“…Denote e := ghg. Since h ∈ M(S), so is e (for M(S) is an ideal of S), and by Theorem 3 of [7] the element e is an idempotent too. Now, ge = g 2 hg = e and eg = ghg 2 = e, hence g ∈ Q(e).…”

Section: Relating Maximal Monoids To Coresmentioning

confidence: 96%

“…Geometrically, it is an (n − 1)-dimensional simplex whose n vertices are the matrices of the form E i = u n (e n i ) T , for i = 1, ..., n. Among the special properties of cores that are of interest to us, stands the fact that every compact affine semigroup S can be expressed as a union of cores of the idempotents in its minimal ideal. The proof of this property, that we write below, makes use of Theorem 3 in the work of Cohen and Collins [7]. x k |n ∈ N} converges to an idempotent e of S such that x ∈ Q(e), as it was proved by H.L.…”

Section: Relating Maximal Monoids To Coresmentioning

confidence: 96%

“…According to Cohen and Collins [7], since this is a group-extremal semigroup, conv H E (I n ) has a zero element, which necessarily is an idempotent Z of Q(E) with rank equal to some positive integer r. This matrix Z is doubly stochastic (for it belongs to the convex hull of a set of permutation matrices) and, in fact, conv H E (I n ) ⊆ Q(E) ∩ D n . But the intersection of these two cores whose zero elements belong to relint M(St n ) must be a core, due to the following consequence of Theorem 3.2.…”

Section: The Maximal Monoids Of Doubly Stochastic Matricesmentioning

confidence: 99%

“…For the basic concepts regarding this type of semigroups we rely on the foundational work of H. Cohen and H.S. Collins [7]; for the concepts used here related to topological semigroups in general, please refer to J.H. Carruth et al [4].…”

Section: Introductionmentioning

confidence: 99%

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A geometric description of the maximal monoids of some matrix semigroups

González-Torres

2015

Linear Algebra and its Applications

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The maximal monoids of the form F SF are studied, where F is a nonnegative idempotent matrix and S is one of the following matrix semigroups: N n , the nonnegative square matrices, St n , the stochastic matrices, and D n , the doubly stochastic matrices. For the cases of N n and St n , it is shown that these maximal monoids are affinely isomorphic to the full semigroups of lower order, and for F D n F that it is a compact affine semigroup with zero, here called the core of a primitive idempotent.

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A note on Haar-like measure for group-extremal semi-groups

Friedberg¹

1966

Proc. Amer. Math. Soc.

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If 5 is a compact group-extremal affine semigroup, it is natural to ask how much of the group structure of the extreme points carries over to S. In [4], the author shows it is possible to extend sufficiently many unitary representations to separate points of S; in the abelian case, this leads to sufficiently many affine semicharacters to separate points of S. We investigate in this note the possibility of extending Haar measure to S. Using results in [l], [2], and [3], it can be seen that a compact affine semigroup which supports an idempotent measure must be of the form XX Y where X and Y are compact convex sets and multiplication is given by (x, y) •(*'> y')~ix, y')-Thus, one cannot hope in general to extend Haar measure and retain all of its properties. However, we will show that if 5 is a compact group-extremal affine semigroup, then there is a probability measure p. supported on 5 (i.e. /^-measure of each nonvoid open set is positive) satisfying

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Affine Semigroups

Cited by 7 publications

References 6 publications

Remarks on affine semigroups

Remarks on affine semigroups

A geometric description of the maximal monoids of some matrix semigroups

A note on Haar-like measure for group-extremal semi-groups

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