Two characterisations of groups amongst monoids

Geometric Characterisations of Groups

Monoids and Groups

1 is unique (1 = 11 = 1, so any left identity equals any right identity). A morphism takes 1 to the identity of φX (since φ(x) = φ(1x) = φ(1)φ(x)), but φ(1) = 1 is an extra property that ought to be satisfied by morphisms; its kernel is the sub-algebra φ(1); zero object is { 1 }; not Cartesian-closed (the terminal and initial objects are the same, yet not all groups are isomorphic). A quasi-gro...

Markov semigroups, monoids, and groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown...

The Semigroup Efficiency of Groups and Monoids

A nite semigroup (respectively monoid or group) S is said to be eecient if it can be deened by a semigroup (respectively monoid or group) presentation hA jRi with jRj ? jAj = rank(H 2 (S 1)). In this paper we show that a group is eecient as a group if and only if it is eecient as a monoid, but some eecient monoids are not eecient as semigroups. Moreover we show that certain classes of eecient g...

Partial symmetry, reflection monoids and Coxeter groups

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.