Representation Extension and Amalgamation in Pomonoids

Abstract

Flatness properties of acts over monoids and their connection with monoid amalgamation have been investigated for almost four decades and a substantial literature on the subject has now appeared. Analogous research, concerning the action of partially ordered monoids on partially ordered sets and its relation to pomonoid amalgamation, was begun in 1980s in two articles by S. M. Fakhruddin. The subject then remained dormant until the recent past when several articles on flatness in the setting of ordered monoids acting on posets (briefly, S-posets) appeared. It has now been established that the introduction of order results in severe restrictions as far as absolute flatness is concerned. Also, after formulating in the ordered context the Representation Extension and Right Congruence Extension Properties, first used by T. E. Hall to study semigroup amalgams, the authors recently observed in their article [4] that inverse monoids, though being amalgamation bases in the class of all monoids, may fail to possess this property when put in the ordered scenario. The purpose of the present article is to formulate an ordered version of the Strong Representation Extension Property of Hall and to explore its connections with absolute flatness and amalgamation of pomonoids. This enables us to prove that pogroups are (strong) amalgamation bases in the class of all pomonoids.

Key Words:

• Absolutely flat
• Amalgamation
• Strong representation extension property
• S-poset
• Ordered monoid

2000 Mathematics Subject Classification:

• Primary 20M30
• Secondary 06F05, 20M50

ACKNOWLEDGMENTS

The authors are grateful to the anonymous referee for his or her careful reading of the manuscript, which resulted in several improvements.

Research of the first-named author is supported by Natural Sciences and Engineering Research Council of Canada Discovery Grant A4494. The second-named author is thankful to the Higher Education Commission, Pakistan, and the Pakistan Navy for their support.

Dedicated to Dr. Barbara Bulman-Fleming.

Notes

Communicated by V. Gould.