Closed subgroups of lattice-ordered permutation groups

Author:
Stephen H. McCleary

Journal:
Trans. Amer. Math. Soc. **173** (1972), 303-314

MSC:
Primary 06A55; Secondary 20B99

MathSciNet review:
0311535

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an -subgroup of the lattice-ordered group of order-preserving permutations of a chain ; and in this abstract, assume for convenience that is transitive. Let denote the completion by Dedekind cuts of . The stabilizer subgroups , will be used to characterize certain subgroups of which are *closed* (under arbitrary suprema which exist in ). If is an *-block* of (a nonempty convex subset such that for any , either or is empty), and if will denote ; and the *-block system* consisting of the translates of will be called *closed* if is closed. When the collection of -block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system , and all systems above are closed. is the *trivial system* (of singletons) iff is *complete* (in ). is closed iff is a *cut in* i.e., is not in the interior of any . Every closed convex -subgroup of is an inter-section of stabilizers of cuts in . Every closed prime subgroup is either a stabilizer of a cut in , or else is minimal and is the intersection of a tower of such stabilizers. is the distributive radical of , so that acts faithfully (and completely) on iff is completely distributive. Every closed -ideal of is for some system . A group in which every nontrivial -block supports some (e.g., a generalized ordered wreath product) fails to be complete iff has a smallest nontrivial system and the restriction is --transitive and lacks elements of bounded support.

These results about permutation groups are used to show that if is an abstract -group having a representing subgroup, its closed -ideals form a tower under inclusion; and that if is a Holland kernel of a completely distributive abstract -group , then so is the set of closures , so that if has a transitive representation as a permutation group, it has a complete transitive representation.

**[1]**Richard D. Byrd,*Complete distributivity in lattice-ordered groups*, Pacific J. Math.**20**(1967), 423–432. MR**0207866****[2]**Richard D. Byrd and Justin T. Lloyd,*Closed subgroups and complete distributivity in lattice-ordered groups*, Math. Z.**101**(1967), 123–130. MR**0218284****[3]**Charles Holland,*The lattice-ordered groups of automorphisms of an ordered set*, Michigan Math. J.**10**(1963), 399–408. MR**0158009****[4]**Charles Holland,*Transitive lattice-ordered permutation groups*, Math. Z.**87**(1965), 420–433. MR**0178052****[5]**W. Charles Holland and Stephen H. McCleary,*Wreath products of ordered permutation groups*, Pacific J. Math.**31**(1969), 703–716. MR**0258704****[6]**Justin T. Lloyd,*Representations of lattice-ordered groups having a basis*, Pacific J. Math.**15**(1965), 1313–1317. MR**0185019****[7]**Justin T. Lloyd,*Complete distributivity in certain infinite permutation groups*, Michigan Math. J.**14**(1967), 393–400. MR**0219462****[8]**Stephen H. McCleary,*Pointwise suprema of order-preserving permutations*, Illinois J. Math.**16**(1972), 69–75. MR**0294202****[9]**Stephen H. McCleary,*The closed prime subgroups of certain ordered permutation groups*, Pacific J. Math.**31**(1969), 745–753. MR**0266833****[10]**Stephen H. McCleary,*0-primitive ordered permutation groups*, Pacific J. Math.**40**(1972), 349–372. MR**0313155****[11]**-, -*-transitive ordered permutation groups*(to appear).**[12]**-,*-primitive ordered permutation groups*, II (to appear).

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1972-0311535-1

Keywords:
Lattice-ordered permutation group,
totally ordered set,
complete subgroup,
prime subgroup,
closed subgroup,
stabilizer subgroup,
complete distributivity,
wreath product

Article copyright:
© Copyright 1972
American Mathematical Society