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Monomorphisms of semigroups of B-rigid local dendrites. (English) Zbl 0615.20043

If X is a topological space, then S(X) denotes the semigroup of continuous maps of X into itself under composition. If \(h: X\to Y\) and \(k: Y\to X\) are such that \(k\circ h: X\to X\) is the identity, then the map S(X)\(\to S(Y)\) defined by \(f\to h\circ f\circ k\) is called a natural map. A local dendrite X is said to be B-rigid if it contains at least one branch point and each homeomorphism from X into X fixes each branch point of X. Let X be any local dendrite, \(p\in X\), and let D be any sub- dendrite of X which is a neighbourhood of p. The number of components of D-\(\{\) \(p\}\) is denoted Rank(p,X) and is called the rank of p in X.
The author’s main theorem states: Let X and Y be local dendrites with finite branch numbers and suppose that X is B-rigid. Then each monomorphism of S(X) into S(Y) is natural if and only if for each copy Z of X contained in Y and each branch point b of Z, \(Rank(b,Z)=Rank(b,Y)\). - There are a number of other related results in this paper including: Let X and Y be spaces. Then there exists a natural monomorphism from S(X) into S(Y) if and only if X is homeomorphic to a retract of Y.
Reviewer: J.Hildebrant

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
20M15 Mappings of semigroups
54H15 Transformation groups and semigroups (topological aspects)
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
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References:

[1] Cook, H.,Continua which admit only the identity mapping onto nondegenerate subcontinua, Fund. Math. 60 (1967) 241–249. · Zbl 0158.41503
[2] Kuratowski, K.,Topology, Vol II, Academic Press, New York and London (1968).
[3] Magill, Jr., K. D.,Semigroups with only finitely many regular D-classes, Semigroup Forum, 25 (1982) 361–377. · Zbl 0499.54031 · doi:10.1007/BF02573610
[4] –,An embedding theorem for semigroups of continuous selfmaps, Semigroup Forum, 27 (1983) 223–236. · Zbl 0536.20041 · doi:10.1007/BF02572740
[5] Magill, Jr., K. D.,Monomorphisms of semigroups of local dendrites, Can. J. Math. (to appear). · Zbl 0589.20049
[6] – and S. Subbiah,Regular J-classes of semigroups of continua, Semigroup Forum, 22 (1981) 159–179. · Zbl 0488.54025 · doi:10.1007/BF02572795
[7] Whyburn, G.,Analytic topology, AMS Colloquium Publication, New York (1942). · Zbl 0061.39301
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