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| In [[mathematics]], the '''bicyclic semigroup''' is an algebraic object important for the structure theory of [[semigroup]]s. Although it is in fact a [[monoid]], it is usually referred to as simply a semigroup.
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| ==History==
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| The first published description of this object was given by [[Evgenii Lyapin]] in 1953. [[Alfred H. Clifford]] and [[Gordon Preston]] claim that one of them, working with [[David Rees (mathematician)|David Rees]], discovered it independently (without publication) at some point before 1943.
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| ==Construction==
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| There are at least three standard ways of constructing the bicyclic semigroup, and various notations for referring to it. Lyapin called it ''P''; Clifford and Preston used <math>\mathcal{C}</math>; and most recent papers have tended to use ''B''. This article will use the modern style throughout.
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| ===From a free semigroup===
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| The bicyclic semigroup is the [[free semigroup]] on two generators ''p'' and ''q'', under the relation ''p'' ''q'' = 1. That is, each semigroup element is a string of those two letters, with the proviso that the subsequence "''p'' ''q''" does not appear.
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| The semigroup operation is concatenation of strings, which is clearly [[associative]].
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| It can then be shown that all elements of ''B'' in fact have the form ''q''<sup>''a''</sup> ''p''<sup>''b''</sup>, for some [[natural number]]s ''a'' and ''b''. The composition operation simplifies to
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| : (''q''<sup>''a''</sup> ''p''<sup>''b''</sup>) (''q''<sup>''c''</sup> ''p''<sup>''d''</sup>) = ''q''<sup>''a'' − ''b'' + max{''b'', ''c''}</sup> ''p''<sup>''d'' − ''c'' + max{''b'', ''c''}</sup>.
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| ===From ordered pairs===
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| The way in which these exponents are constrained suggests that the "''p'' and ''q'' structure" can be discarded, leaving only operations on the "''a'' and ''b''" part.
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| So ''B'' is the semigroup of pairs of natural numbers (including zero), with operation<ref>Hollings (2007), p. 332</ref>
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| :(''a'', ''b'') (''c'', ''d'') = (''a'' − ''b'' + max{''b'', ''c''}, ''d'' − ''c'' + max{''b'', ''c''}).
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| This is sufficient to define ''B'' so that it is the same object as in the original construction. Just as ''p'' and ''q'' generated ''B'' originally, with the empty string as the monoid identity, this new construction of ''B'' has generators (1, 0) and (0, 1), with identity (0, 0).
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| ===From functions===
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| It can be shown that ''any'' semigroup ''S'' generated by elements ''e'', ''a'', and ''b'' satisfying the statements below is [[Isomorphism|isomorphic]] to the bicyclic semigroup.
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| * ''a'' ''e'' = ''e'' ''a'' = ''a''
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| * ''b'' ''e'' = ''e'' ''b'' = ''b''
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| * ''a'' ''b'' = ''e''
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| * ''b'' ''a'' ≠ ''e''
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| It is not entirely obvious that this should be the case—perhaps the hardest task is understanding that ''S'' must be infinite. To see this, suppose that ''a'' (say) does not have infinite order, so ''a''<sup>''k'' + ''h''</sup> = ''a''<sup>''h''</sup> for some ''h'' and ''k''. Then ''a''<sup>''k''</sup> = ''e'', and
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| :''b'' = ''e'' ''b'' = ''a''<sup>''k''</sup> ''b'' = ''a''<sup>''k'' - 1</sup> ''e'' = ''a''<sup>''k'' - 1</sup>,
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| so
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| :''b'' ''a'' = ''a''<sup>''k''</sup> = ''e'',
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| which is not allowed—so there are infinitely many distinct powers of ''a''. The full proof is given in Clifford and Preston's book.
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| Note that the two definitions given above both satisfy these properties. A third way of deriving ''B'' uses two appropriately-chosen functions to yield the bicyclic semigroup as a monoid of transformations of the natural numbers. Let α, β, and ι be elements of the [[transformation semigroup]] on the natural numbers, where
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| *ι(''n'') = ''n''
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| *α(''n'') = ''n'' + 1
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| *β(''n'') = 0 if ''n'' = 0, and ''n'' − 1 otherwise.
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| These three functions have the required properties, so the semigroup they generate is ''B''.<ref name=Lot459>{{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 | page=459 }}</ref>
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| ==Properties==
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| The bicyclic semigroup has the property that the image of any [[morphism]] φ from ''B'' to another semigroup ''S'' is either cyclic, or it is an isomorphic copy of ''B''. The elements φ(''a''), φ(''b'') and φ(''e'') of ''S'' will always satisfy the conditions above (because φ is a morphism) with the possible exception that φ(''b'') φ(''a'') might turn out to be φ(''e''). If this is not true, then φ(''B'') is isomorphic to ''B''; otherwise, it is the cyclic semigroup generated by φ(''a''). In practice, this means that the bicyclic semigroup can be found in many different contexts.
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| The [[idempotent]]s of ''B'' are all pairs (''x'', ''x''), where ''x'' is any natural number (using the ordered pair characterisation of ''B''). Since these commute, and ''B'' is ''regular'' (for every ''x'' there is a ''y'' such that ''x'' ''y'' ''x'' = ''x''), the bicyclic semigroup is an [[inverse semigroup]]. (This means that each element ''x'' of ''B'' has a unique inverse ''y'', in the "weak" semigroup sense that ''x'' ''y'' ''x'' = ''x'' and ''y'' ''x'' ''y'' = ''y''.) | |
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| Every [[Ideal (ring theory)|ideal]] of ''B'' is principal: the left and right principal ideals of (''m'', ''n'') are
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| * (''m'', ''n'') ''B'' = {(''s'', ''t'') : ''s'' ≥ ''m''} and
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| * ''B'' (''m'', ''n'') = {(''s'', ''t'') : ''t'' ≥ ''n''}.
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| Each of these contains infinitely many others, so ''B'' does not have minimal left or right ideals.
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| In terms of [[Green's relations]], ''B'' has only one ''D''-class (it is ''bisimple''), and hence has only one ''J''-class (it is ''simple''). The ''L'' and ''R'' relations are given by
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| * (''a'', ''b'') ''R'' (''c'', ''d'') [[if and only if]] ''a'' = ''c''; and
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| * (''a'', ''b'') ''L'' (''c'', ''d'') if and only if ''b'' = ''d''.<ref>Howie p.60</ref>
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| This implies that two elements are ''H''-related if and only if they are identical. Consequently, the only subgroups of ''B'' are infinitely many copies of the trivial group, each corresponding to one of the idempotents.
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| The [[Green's relations|egg-box diagram]] for ''B'' is infinitely large; the upper left corner begins:
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| {| cellpadding=6
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| | (0, 0) || (1, 0) || (2, 0) || ...
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| |-
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| | (0, 1) || (1, 1) || (2, 1) || ...
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| |-
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| | (0, 2) || (1, 2) || (2, 2) || ...
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| |-
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| | ... || ... || ... || ...
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| |}
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| Each entry represents a singleton ''H''-class; the rows are the ''R''-classes and the columns are ''L''-classes. The idempotents of ''B'' appear down the diagonal, in accordance with the fact that in a regular semigroup with commuting idempotents, each ''L''-class and each ''R''-class must contain exactly one idempotent.
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| The bicyclic semigroup is the "simplest" example of a bisimple inverse semigroup with identity; there are many others. Where the definition of ''B'' from ordered pairs used the class of natural numbers (which is not only an additive semigroup, but also a commutative [[Lattice (order)|lattice]] under min and max operations), another set with appropriate properties could appear instead, and the "+", "−" and "max" operations modified accordingly.
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| ==Relation to combinatorics==
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| The bicyclic monoid occurs in [[combinatorics]], as the [[syntactic monoid]] of the [[Dyck language]]. The Dyck language is the set of all strings of balanced pairs of parentheses, and thus finds common applications in defining [[binary tree]]s and [[associative algebra]]s.
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| ==See also==
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| *[[Four-spiral semigroup]]
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| *[[Special classes of semigroups]]
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| == Notes ==
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| <references />
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| ==References==
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| * ''The algebraic theory of semigroups'', A. H. Clifford and G. B. Preston. American Mathematical Society, 1961 (volume 1), 1967 (volume 2).
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| * ''Semigroups: an introduction to the structure theory'', Pierre Antoine Grillet. Marcel Dekker, Inc., 1995.
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| * ''Canonical form of elements of an associative system given by defining relations'', Evgenii Sergeevich Lyapin, ''Leningrad Gos. Ped. Inst. Uch. Zap.'' '''89''' (1953), pages 45–54 [Russian].
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| * {{cite journal |last1=Hollings |first1=C.D. |last2= |first2= |year=2007 |title=Some First Tantalizing Steps into Semigroup Theory |journal=Mathematics Magazine |volume=80 |issue= |pages=331–344 |publisher=Mathematical Association of America |doi= |jstor=27643058 }}
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| [[Category:Semigroup theory]]
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