Transitive partial actions of groups
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TRANSITIVE PARTIAL ACTIONS OF GROUPS Keunbae Choi1 and Yongdo Lim2 [Communicated by M´ aria B. Szendrei] 1
Department of Mathematics Education, Teachers College, Cheju National University Jeju 690-781, Korea E-mail: [email protected] 2
Department of Mathematics, Kyungpook National University Taegu 702-701, Korea E-mail: [email protected] (Received April 29, 2003; Accepted December 18, 2007)
Abstract J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set YG containing a copy of Y . In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on YG such that the global group action of G on YG is continuous and the injection Y into YG is an open dense equivariant embedding.
1. Introduction Throughout this paper we shall always assume that G is a group with multiplication gh (g, h ∈ G). 1 denotes the identity of G and g −1 denotes the inverse of g in G. Recall that an action of G on the set X is a function G×X → X, (g, x) → g ·x such that 1 · x = x, g · (h · x) = (gh) · x for all g, h ∈ G and x ∈ G, and that it can also be defined by means of a homomorphism from G to the symmetric group on X. Two G-actions on X and X are said to be equivalent if there is a bijection f : X → X such that f (g · x) = g · f (x) for all g ∈ G and x ∈ X. Such a map f is called an isomorphism between two G-actions. An inverse semigroup S is a semigroup in which for every s ∈ S there exists a unique element s−1 , called the inverse of s, satisfying ss−1 s = s, s−1 ss−1 = s−1 . The Wagner–Preston representation theorem ([6], [9]) states that every inverse monoid can be embedded in a symmetric inverse monoid I(X) on a set X: this Mathematics subject classification number : 54H15, 20M18. Key words and phrases: partial action, symmetric inverse monoid. 0031-5303/2008/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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K. CHOI and Y. LIM
consists of all partial bijections on the set X under the usual operation of composition of partial functions. Let ≤ be the natural partial order on the inverse monoid I(X). Then for s, t ∈ I(X), s ≤ t if and only if the domain of t contains the domain of s and the two partial maps s and t agree on the domain of s. A partial action (this terminology was introduced by R. Exel, [7]) of G on a non-empty set Y is a function θ: G → I(Y ) satisfying the following three conditions: (P1) θ(g −1 ) = θ(g)−1 for all g ∈ G; (P2) θ(g)θ(h) ≤ θ(gh) for all g, h ∈ G; (P3) θ(1) = 1Y , the identity on Y . Notice that the essential difference between partial group actions and group actions lies in condition (P2): θ(gh) is an extension of θ(g)θ(h) for all g, h ∈ G. For an action of G on a set X, G × X → X, (g, x) → g · x, and a non-empty subset Y ⊂ X, each element of g ∈ G induces a partial bijection of Y whose domain is given by {y ∈ Y : g · y ∈ Y }, and hence the
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