Four-spiral Semigroup
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the four-spiral semigroup is a special
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
generated by four
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
elements. This special semigroup was first studied by Karl Byleen in a doctoral dissertation submitted to the
University of Nebraska A university () is an educational institution, institution of tertiary education and research which awards academic degrees in several Discipline (academia), academic disciplines. ''University'' is derived from the Latin phrase , which roughly ...
in 1977. It has several interesting properties: it is one of the most important examples of bi-simple but not completely-simple semigroups; it is also an important example of a fundamental
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
; it is an indispensable building block of bisimple, idempotent-generated regular semigroups. A certain semigroup, called double four-spiral semigroup, generated by five idempotent elements has also been studied along with the four-spiral semigroup.


Definition

The four-spiral semigroup, denoted by ''Sp4'', is the
free semigroup In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ...
generated by four elements ''a'', ''b'', ''c'', and ''d'' satisfying the following eleven conditions: :* ''a''2 = ''a'', ''b''2 = ''b'', ''c''2 = ''c'', ''d''2 = ''d''. :* ''ab'' = ''b'', ''ba'' = ''a'', ''bc'' = ''b'', ''cb'' = ''c'', ''cd'' = ''d'', ''dc'' = ''c''. :* ''da'' = ''d''. The first set of conditions imply that the elements ''a'', ''b'', ''c'', ''d'' are idempotents. The second set of conditions imply that ''a R b L c R d'' where ''R'' and ''L'' are the
Green's relations In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. ...
in a semigroup. The lone condition in the third set can be written as ''d'' ωl ''a'', where ωl is a biorder relation defined by Nambooripad. The diagram below summarises the various relations among ''a'', ''b'', ''c'', ''d'': \begin & & \mathcal & & \\ & a & \longleftrightarrow & b & \\ \omega^l & \Big \uparrow & & \Big \updownarrow & \mathcal \\ & d & \longleftrightarrow & c & \\ & & \mathcal & & \end


Elements of the four-spiral semigroup


General elements

Every element of ''Sp''4 can be written uniquely in one of the following forms: :: 'c''(''ac'')''m'' :: 'd''(''bd'')''n'' 'b'':: 'c''(''ac'')''m'' ''ad'' (''bd'')''n'' 'b''where ''m'' and ''n'' are non-negative integers and terms in square brackets may be omitted as long as the remaining product is not empty. The forms of these elements imply that ''Sp''4 has a partition ''Sp''4 = ''A'' ∪ ''B'' ∪ ''C'' ∪ ''D'' ∪ ''E'' where :: ''A'' = :: ''B'' = :: ''C'' = :: ''D'' = :: ''E'' = The sets ''A'', ''B'', ''C'', ''D'' are
bicyclic semigroup 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 usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic m ...
s, ''E'' is an infinite
cyclic semigroup In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups. Structure The monogenic semigroup generated by the singleton set is denoted by \langle a \rangle. The set ...
and the subsemigroup ''D'' ∪ ''E'' is a nonregular semigroup.


Idempotent elements

The set of idempotents of ''Sp''4, is where, ''a''0 = ''a'', ''b''0 = ''b'', ''c''0 = ''c'', ''d''0 = ''d'', and for ''n'' = 0, 1, 2, ...., :: ''a''''n''+1 = ''a''(''ca'')''n''(''db'')''n''''d'' :: ''b''''n''+1 = ''a''(''ca'')''n''(''db'')''n''+1 :: ''c''''n''+1 = (''ca'')''n''+1(''db'')''n''+1 :: ''d''''n''+1 = (''ca'')''n''+1(''db'')''n''+l''d'' The sets of idempotents in the subsemigroups ''A'', ''B'', ''C'', ''D'' (there are no idempotents in the subsemigoup ''E'') are respectively: :: ''E''''A'' = :: ''E''''B'' = :: ''E''''C'' = :: ''E''''D'' =


Four-spiral semigroup as a Rees-matrix semigroup

Let ''S'' be the set of all quadruples (''r'', ''x'', ''y'', ''s'') where ''r'', ''s'', ∈ and ''x'' and ''y'' are nonnegative integers and define a binary operation in ''S'' by (r, x, y, s) * (t, z, w, u) = \begin (r, x-y + \max(y , z + 1), \max(y - 1, z) - z + w, u) & \text s = 0, t = 1\\ (r, x - y+ \max(y, z), \max(y, z) - z + w, u)&\text \end The set ''S'' with this operation is a
Rees matrix semigroup Rees may refer to: Places * Rees, Germany Rees () is a town in the Kleve (district), district of Kleve in the state of North Rhine-Westphalia, Germany. It is located on the right bank of the Rhine, approximately 20 km east of Kleve. The p ...
over the
bicyclic semigroup 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 usually referred to as simply a semigroup. It is perhaps most easily understood as the syntactic m ...
, and the four-spiral semigroup ''Sp''4 is isomorphic to ''S''.


Properties

*By definition itself, the four-spiral semigroup is an ''idempotent generated semigroup'' (''Sp''4 is generated by the four idempotents ''a'', ''b''. ''c'', ''d''.) *The four-spiral semigroup is a fundamental semigroup, that is, the only congruence on ''Sp''4 which is contained in the Green's relation ''H'' in ''Sp''4 is the equality relation.


Double four-spiral semigroup

The fundamental double four-spiral semigroup, denoted by ''DSp''4, is the semigroup generated by five elements ''a'', ''b'', ''c'', ''d'', ''e'' satisfying the following conditions: :*''a''2 = ''a'', ''b''2 = ''b'', ''c''2 = ''c'', ''d''2 = ''d'', ''e''2 = ''e'' :*''ab'' = ''b'', ''ba'' = ''a'', ''bc'' = ''b'', ''cb'' = ''c'', ''cd'' = ''d'', ''dc'' = ''c'', ''de'' = ''d'', ''ed'' = ''e'' :*''ae'' = ''e'', ''ea'' = ''e'' The first set of conditions imply that the elements ''a'', ''b'', ''c'', ''d'', ''e'' are idempotents. The second set of conditions state the Green's relations among these idempotents, namely, ''a R b L c R d L e''. The two conditions in the third set imply that ''e'' ω ''a'' where ω is the biorder relation defined as ω = ωl ∩ ωr.


References

{{reflist Semigroup theory 4 (number) Spirals