Semigroup - Special Classes of Semigroups

Special Classes of Semigroups

• A monoid is a semigroup with identity.
• A subsemigroup is a subset of a semigroup that is closed under the semigroup operation.
• A band is a semigroup the operation of which is idempotent.
• A cancellative semigroup is one having the cancellation property: a · b = a · c implies b = c and similarly for b · a = c · a.
• A semilattice is a semigroup whose operation is idempotent and commutative.
• 0-simple semigroups.
• Transformation semigroups: any finite semigroup S can be represented by transformations of a (state-) set Q of at most |S|+1 states. Each element x of S then maps Q into itself x: Q → Q and sequence xy is defined by q(xy) = (qx)y for each q in Q. Sequencing clearly is an associative operation, here equivalent to function composition. This representation is basic for any automaton or finite state machine (FSM).
• The bicyclic semigroup is in fact a monoid, which can be described as the free semigroup on two generators p and q, under the relation p q = 1.
• C₀-semigroups.
• Regular semigroups. Every element x has at least one inverse y satisfying xyx=x and yxy=y; the elements x and y are sometimes called "mutually inverse".
• Inverse semigroups are regular semigroups where every element has exactly one inverse. Alternatively, a regular semigroup is inverse if and only if any two idempotents commute.
• Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Zd. These semigroups have applications to commutative algebra.