Categorical semigroups

On 0-homology of categorical at zero semigroups

The isomorphism of 0-homology groups of a categorical at zero semigroup and homology groups of its 0-reflector is proved. Some applications of 0-homology to Eilenberg–MacLane homology of semigroups are given.

Automatic semigroups vs automaton semigroups

We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup, namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication. The framework embraces many of the well-known classes of (automatic) semigroups: free semigroups,...

Categorical Logic Categorical Logic

Finite semigroups as categories, ordered semigroups or compact semigroups

The results presented in this section are a good illustration of the following quotation of Marshall Stone [34]: ’A cardinal principle of modern mathematical research may be stated as a maxim: “One must always topologize” ’. Varieties of finite semigroups are a good example where Stone’s principle was applied successfully. Recall that a variety of semigroups is a class of semigroups closed unde...

Categorical representations of categorical groups

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail.