Applications of the Cayley graph of a monoid presentation in characterizations for right type B semigroups

Motivated by the rich results on Cayley graphs of semigroups, in this paper we investigate the Cayley graph of a monoid presentation, and get a method which makes it possible to construct a certain type of a monoid (which we call graph expansions) by the Cayley graph of a monoid presentation. After obtaining some basic properties of Cayley graphs of a monoid presentation, we give a new characterization of a right type B semigroup by using a Cayley graph of a monoid presentation of a left cancellative monoid. It is proved that an arbitrary graph expansion for a left cancellative monoid is a proper (F) right type B monoid. AMS Mathematics Subject Classification (2010) 05C25; 20M10