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In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is
Bicyclic_semigroup
Algebraic structure
appears in the theory of one-parameter operator semigroups: see C0-semigroup. The binary operation of a semigroup is most often denoted multiplicatively: x
Semigroup
meant by a regular band. The bicyclic semigroup is regular. Any full transformation semigroup is regular. A Rees matrix semigroup is regular. The homomorphic
Regular_semigroup
Families of certain algebraic structures
mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying
Special_classes_of_semigroups
Structure in group theory (in mathematics)
semigroup S. Partial bijections on a set X form an inverse semigroup under composition. Every group is an inverse semigroup. The bicyclic semigroup is
Inverse_semigroup
Algebraic structure in mathematics
matrix semigroup over the bicyclic semigroup, and the four-spiral semigroup Sp4 is isomorphic to S. By definition itself, the four-spiral semigroup is an
Four-spiral_semigroup
a finite set of generators is compact. The bicyclic monoid is not compact. The class of compact semigroups is closed under taking subsemigroups and finite
Compact_semigroup
Mathematical structure
classes of semigroups, notably completely simple semigroups (Campbell et al. 2002) and group-embeddable semigroups (Cain et al. 2006). Bicyclic monoid Finitely
Automatic_semigroup
Language consisting of balanced strings of brackets
. The syntactic monoid of the Dyck language is isomorphic to the bicyclic semigroup by virtue of the properties of Cl ( [ ) {\displaystyle \operatorname
Dyck_language
presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set Σ of generators and a set of
Presentation_of_a_monoid
found for example in the bicyclic semigroup, is where each element is in an H-class of its own. The egg-box for this semigroup would contain infinitely
Green's_relations
Algebraic structure with an associative operation and an identity element
with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of
Monoid
Smallest monoid that recognizes a formal language
minimal automaton has 4 states and the syntactic monoid has 15 elements. The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced
Syntactic_monoid
String rewriting system
introduced this notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable— this result was
Semi-Thue_system
South African Canadian computer scientist (1918–2013)
Doctor of Philosophy (Ph.D.) in 1950, with a thesis on the topological semigroups. He then went on to teach at Brown University for three years before returning
John_E._L._Peck
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