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Semigroups of transformations restricted by an equivalence

100%

Mendes-Gonçalves S., Sullivan R.

Open Mathematics

2010

tom 8

nr 6

1120-1131

Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.

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Correspondences between ideals and \(z\)-filters for rings of continuous functions between \(C^∗\) and \(C\)

88%

Panman P., Sack J., Watson S.

Commentationes Mathematicae

2012

tom 52

nr 1

Let \(X\) be a completely regular topological space. Let \(A(X)\) be a ring of continuous functions between \(C^∗(X)\) and \(C(X)\), that is, \(C^∗(X) \subset A(X) \subset C(X)\). In [9], a correspondence \(\mathcal{Z}_A\) between ideals of \(A(X)\) and \(z\)-filters on \(X\) is defined. Here we show that \(\mathcal{Z}_A\) extends the well-known correspondence for \(C^∗(X)\) to all rings \(A(X)\). We define a new correspondence \(\mathcal{Z}_A\) and show that it extends the well-known correspondence for \(C(X)\) to all rings \(A(X)\). We give a formula that relates the two correspondences. We use properties of \(\mathcal{Z}_A\) and \(\mathcal{Z}_A\) to characterize \(C^∗(X)\) and \(C(X)\) among all rings \(A(X)\). We show that \(\mathcal{Z}_A\) defines a one-one correspondence between maximal ideals in \(A(X)\) and the \(z\)-ultrafilters in \(X\).

Ograniczanie wyników

1 Commentationes Mathematicae

1 Open Mathematics

1 Mendes-Gonçalves S.

1 Panman P.

1 Sack J.

1 Sullivan R.

1 Watson S.

1 2012

1 2010

Wyniki wyszukiwania

Semigroups of transformations restricted by an equivalence

Correspondences between ideals and \(z\)-filters for rings of continuous functions between \(C^∗\) and \(C\)