Near heaps
Ian Hawthorn, Tim Stokes
Abstrat.On any involuted semigroup (S;; 0
), dene the ternary operation
[ab℄:=ab 0
foralla;b;2S. Theresultingternaryalgebra(S;[℄)satises
thepara-assoiativitylaw[[ab℄de℄=[a[db℄e℄=[ab[de℄℄,whihdenesthevari-
etyofsemiheaps. Importantsubvarietiesinludegeneralisedheaps,whiharise
frominversesemigroups,andheaps,whiharisefromgroups. Weonsiderthe
intermediatevarietyofnearheaps,denedbytheadditionallaws[aaa℄=aand
[aab℄=[baa℄. EveryCliordsemigroupisanearheapwhenviewedasa semi-
heap,andweshowthattheCliordsemigroupoperationsaredeterminedbythe
semiheapoperation. Weshowthatnearheapsareexatlystrongsemilattiesof
heaps,parallellingaknownresultforCliordsemigroups.Weharaterisethose
nearheapswhiharisediretlyfromCliordsemigroups,andshowthatallnear
heaps areembeddable insuhexamples, extendingknownresults of thiskind
relating heapstogroups, generalisedheapstoinversesemigroups, andgeneral
semiheapstoinvolutedsemigroups.
Keywords:Cliordsemigroups,semiheaps,generalisedheaps,heaps
Classiation: Primary20N10;Seondary20M11
1. Bakground on semiheaps
Webeginwithareviewofsomeestablisheddenitions andresults.
Aheap H isanon-emptysetwithternaryoperation[℄satisfyingthefollowing
laws.
[[ab℄de℄=[a[db℄e℄=[ab[de℄℄(para-assoiativelaw)
[aab℄=[baa℄=b
Weall theone-element heapthe trivialheap. Everygroupgivesaheap under
theternaryoperation[ab℄:=ab 1
,aonstrutionrstonsideredinthesetting
ofabeliangroupsbyPruferin [4℄. Conversely, agrouparises fromaheapH by
hoosing anyelement e2H and dening abinary operationxy :=[xey℄; the
elementebeomestheidentityoftheonstrutedgroupand[exe℄theinverseofx.
Theseonstrutionsaremutuallyinverseuptoisomorphism. Henethevarieties
ofgroupsandpointedheapsaretermequivalent,asshownbyBaerin [1℄.
A semiheap H isanon-empty setwith aternaryoperation[℄ satisfyingonly
thepara-assoiativelawabove. SemiheapswererstonsideredbyWagnerin[5℄.
A similaronstrutiongivesasemiheap whenS isan involuted semigroup, that
isasemigroupequippedwithaunaryoperation 0
forwhihthefollowinglawsare
a 00
=a
(ab) 0
=b 0
a 0
.
If S is an involuted semigroup, setting [ab℄ := ab 0
for all a;b; 2 S gives a
semiheap operation onS. Denote by [S℄ the semiheap obtainedfrom S in this
way. Everysemiheapanbeembeddedin [S℄forsomeinvolutedsemigroupS: see
Setion2of[5℄.
Anidempotent semiheap isasemiheapsatisfying
[aaa℄=a(idempotenylaw).
These were studied in [2℄. If S is an involuted semigroup, the semiheap [S℄ is
idempotent ifand only if aa 0
a= a (see [3℄). An involutedsemigroup with this
property isalled aninvolutedI-semigroup. Everyidempotentsemiheap anbe
embedded in [S℄forsomeinvolutedI-semigroupS. (Thisspei resultdoesnot
appearintheliterature,buttheproofoftheanalogousresultforgeneralisedheaps
in[5℄iseasilymodiedto showit.)
Ageneralisedheap isanidempotentsemiheapsatisfying
[aa[bb℄℄=[bb[aa℄℄and[[abb℄℄=[[a℄bb℄ (generalisedheapaxiom).
These were onsidered byWagner in [5℄. They arise naturallyin the setting of
atlases in dierential geometry; see [6℄. An inverse semigroup is an involuted
semigroupsatisfying
aa 0
a=a(idempoteny)
aa 0
bb 0
=bb 0
aa 0
Omittingthelaw(ab) 0
=b 0
a 0
,whih followsfrom theothers, givesHowie'sde-
nition asonpage 145of [3℄. Itan be shown that theset of idempotents ofan
inversesemigroupS isE(S)=fa 0
aja2Sgsothat idempotentsommuteinan
inverse semigroup. If S isan involutedsemigroup, thesemiheap [S℄ is agener-
alisedheapifanonlyifS is aninversesemigroup,andallgeneralisedheapsan
beembeddedinageneralisedheaponstrutedinthis way(seeSetion3of[5℄).
Wenowsummarizetheseresults.
Proposition 1. LetS beaninvolutedsemigroup.
(1) [S℄isasemiheap. Everysemiheapanbeembeddedinasemiheapofthis
type.
(2) [S℄is anidempotentsemiheapifandonlyif S isanI-semigroup. Every
idempotentsemiheapanbeembeddedinanidempotentsemiheapofthis
type.
(3) [S℄is ageneralisedheapifand onlyif S isaninversesemigroup. Every
generalisedheapan beembeddedin ageneralisedheapof thistype.
(4) [S℄isaheapifandonlyif S isagroup. Allheapsareofthistype.
The orrespondenes an be tightened further, to resemble the situation for
heapsand groups, ifa further assumption is made. If H is asemiheap, wesay
only ifeveryelement is bi-unitary. Note that if S is an involuted monoid with
identity1,then12[S℄isbi-unitary.
Callasemiheapequippedwithadistinguishedbi-unitaryelemente,viewedas
anullaryoperation, bi-unital. The lassofbi-unital semiheapsisavarietywith
oneternaryandonenullaryoperation.
LetH beabi-unitalsemiheap. Dene x 0
=[exe℄forall x2H. Then x 00
=x
forallx2H asiseasilyheked. AlsodeneabinaryoperationonH bysetting
xy =[xey℄. CalltheresultingalgebrahHi. InfathHiisaninvolutedmonoid
withidentitye,asis showninSetion 2of[5℄.
Theonstrutions S 7! [S℄ (S an involutedmonoid) andH 7!hHi(H a bi-
unitalsemiheap)aremutuallyinverse,leadingtoatermequivalenebetweenthe
varietiesofinvolutedmonoidsand bi-unitalsemiheaps. Indeedtheterm equiva-
lenespeialisestosubvarietiesassummarisedin thefollowing.
Proposition 2. The following subvarieties of involuted monoids and bi-unital
semiheapsaretermequivalent.
bi-unitalsemiheaptype involutedmonoidtype
arbitrary arbitrary
idempotentsemiheap I-monoid
generalisedheap inversemonoid
heap group
Eahoftheseequivalenesisestablishedin[5℄(orfollowseasilyfromarguments
giventhere).
2. Introduingnear heaps
Historially, heaps were the rst type of semiheaps onsidered. There are
variouswaystoweakentheheapaxioms,andwehavereviewedthemostimportant
onesalready. Another,rstintroduedin[2℄,givesrisetothelassofnearheaps.
Anearheap isasemiheapsatisfyingthelaws
[aaa℄=a(idempotentlaw)
[aab℄=[baa℄(nearheapaxiom).
Everyheapisanearheapandeverynearheapisageneralisedheap(seePropo-
sition15of [2℄).
LetE(S)=faa 0
ja2Sgdenotethesemilattieofidempotentsin theinverse
semigroup S, partially ordered via e f if and only if e = ef. A Cliord
semigroupSisaninversesemigroupinwhihforalla2Sande2E(S),ae=ea.
Itiswellknown thatthis onditionis equivalenttotheonditionthat aa 0
=a 0
a
for all a2 S (see [3℄ for example). It is lear that ifS is aCliord semigroup
then[S℄isanearheap. Theonverseisalsotrue.
Proposition3. LetSbeaninvolutedsemigroup. Then[S℄isanearheapifand
Proof: IfS is aCliord semigroupthen[S℄ isasemiheap,and ifa;b2S then
[aab℄=aa 0
b=baa 0
=ba 0
a=[baa℄so[S℄is anearheapaslaimed.
Converselyassumethat[S℄isanearheap. ThenS isaninversesemigroupby
Proposition1. Thenforalla2S,letting aa 0
=e2E(S)anda 0
a=f 2E(S)we
have
aa 0
=(aa 0
)(aa 0
)=aa 0
e=[aae℄=[eaa℄=ea 0
a=ef =fe
a 0
a=(a 0
a)(a 0
a)=fa 0
a=[faa℄=[aaf℄=aa 0
f =ef =fe
soaa 0
=a 0
aandS isaCliordsemigroup.
Sothersthalf ofanewentryin Proposition 1annowbemade. A further
entry analso beadded, byspeialisingnear heaps in adiretion orthogonal to
heaps.
SemilattiesmaybethoughtofasCliordsemigroupsinwhihthelawa=a 0
holds; byuniqueness of inverses in inverse semigroups, this is the only possible
waytodene 0
toyieldaCliordsemigroup.
SoifS isasemilattie,thesemiheap operationon[S℄ isgivenby[ab℄=ab
for all a;b; 2 S. The resulting near heap satises thelaw [abb℄ =[aab℄. The
followingeasyresultwasstatedin [2℄(andmayhaverstappearedearlier).
Proposition 4. Let S beaninvolutedsemigroup. [S℄is anearheapsatisfying
thelaw[abb℄=[aab℄ifand onlyif S isasemilattie. All suh nearheapsareof
thistype,andthetwovarietiesaretermequivalent.
Fromnowon,whenweusethetermsemilattieintheontextofsemiheaps,we
meananearheapsatisfyingthelaw[abb℄=[aab℄. Notethat theonlysemilattie
whih isaheapisthetrivialheap,sineitsatisesa=[abb℄=[aab℄=b foralla
andb.
Anewrowin thetableprovidedin Proposition2analsobegiven.
Proposition5. ThevarietyofCliordmonoidsistermequivalenttothevariety
ofbi-unitalnear heaps.
Proof: ItsuÆestoshowthatifH isanearheapwithdistinguishedbi-unitary
elemente,thenhHiisaCliordmonoid(withidentitye). ButbeauseHisanear
heap,itisageneralisedheap,sohHiisaninversesemigroupbywhatisshownin
Setion3of[5℄,andforanyx;y,[xxy℄=[yxx℄soxx 0
y=yx 0
x,solettingy=xx 0
gives(xx 0
) 2
=(xx 0
)(x 0
x), soin thesemilattie E(S), xx 0
x 0
x, andsymmetry
establishestheequalityneededto showthathHiisaCliordsemigroup.
However, in the absene of a bi-unitary element, the orrespondene breaks
down,aswenextshow.
3. Whihnear heapsare [S℄ forsomeS?
Let H be a near heap. Let be the binary relation on H given by a b
if [abb℄ = a and [baa℄ = b. The following is shown in [2℄: see Proposition 6,
Proposition 6. LetH beanearheap. ThenisaongrueneonH forwhih
H=isasemilattie,and foreverya2H, theongruenelass ontainingais
asubsemiheap of H whih isamaximalsubheap.
FromProposition16of[2℄, asemilattieanontainnonon-trivialsubheaps,
this ongrueneanberegardedasdening the`heap radial' ofthenear heap;
see[2℄forfurtherdisussionofradialsofsemiheaps.
IfH isaneapheapweall H=theassoiatedsemilatttie.
Proposition7. If SisaCliordsemigroup,theneveryelementof Sisongruent
undertoauniqueelementof E(S).
Proof: Nowaaa 0
2E(S)sine
[aa(aa 0
)℄=aa 0
aa 0
=aa 0
[(aa 0
)aa℄=aa 0
a 0
a=aa 0
[(aa 0
)(aa 0
)a℄=aa 0
aa 0
a=a
[a(aa 0
)(aa 0
)℄=aaa 0
aa 0
=a:
Also aa 0
bb 0
implies aa 0
= [(aa 0
)(bb 0
)(bb 0
)℄ = aa 0
bb 0
=[(aa 0
)(aa 0
)(bb 0
)℄ = bb 0
,
provinguniqueness.
TheongruenemapsE(S)bijetivelyontotheassoiatedsemilattie[S℄=.
Itfollowsthatin nearheapsoftheform[S℄whereS isaCliord semigroup,the
anonialhomomorphism [S℄ ! [S℄= issplit (hasarightinverse [S℄= !
[S℄). Equivalentlyinthesenearheapsthereisasubsemilattie,inthisaseE(S),
withanelementineveryongruenelassof.
Wedeneaspine foranear heapH to beasubsemilattie Lof H suh that
everyelementh 2 H is ongruent under to a uniqueelement of L. Hene a
neessaryonditionforanearheapH tobe(isomorphito)[S℄forsomeCliord
semigroupisthat ithasaspine.
Thisonditionisnotonlyneessarybutalso suÆientaswenowshow.
Theorem 8. A near heapH is equalto [S℄ forsomeCliord semigroupifand
onlyifithasaspine.
Proof: AsE(S)isaspineof[S℄,theonditionisneessary.
ToshowthatitissuÆientletH beanear heapwith spineLH. Foreah
element h2 H there is auniquee
h
2L with e
h
h. Dene a 0
= [e
a ae
a
℄ and
ab=[ae
a b℄=[ae
b
b℄. Welaimthat(H;; 0
)isaCliordsemigroupunderthese
operationsandthat[(H;; 0
)℄=H.
Firstnotethat
[ae
a
b℄=[ae
a [e
b e
b
b℄℄=[a[e
b e
b e
a
℄b℄=[a[e
b e
a e
a
℄b℄=[[ae
a e
a
℄e
b b℄=[ae
b b℄
sotheprodutasstatedaboveiswelldened.Furthermore(ab)=[[ae
a b℄e
℄=
[ae
a [be
℄℄=a(b)sotheprodutisassoiative.
Nowleta2Handlete=e
a
tosimplifythenotation. Then[eea 0
℄=[ee[eae℄℄=
0 0 0
[eee℄=e. Henea 0
e,whihgivese
a 0
=e
a
=eanda 00
=[e[eae℄e℄=a. Wealso
haveaa 0
a =[[ae[eae℄℄ea℄=a. Furthermore aa 0
=[ae[eae℄℄ =[a[aee℄e℄ =
[aae℄=eso(aa 0
)b =eb=[eeb℄=[bee℄=b(aa 0
)provingthe Cliord
semigroupondition.
At this point we have proved that H is a Cliord semigroup (the ondition
(ab) 0
=b 0
a 0
followsfrom theothers). It remainsto showthat thesemiheap
produt anbereoveredfrom theCliord semigroupoperations. But ab 0
=
[ae
b 0
b 0
℄=[ae
b [e
b be
b
℄℄=[abe
b
℄andhene
ab 0
=[[abe
b
℄e
℄=[a[e
e
b
b℄℄=[a[e
[e
b e
b e
b
℄b℄℄=[a[[e
e
b e
b
℄e
b b℄℄
=[a[[e
e
e
b
℄e
b
b℄℄=[a[e
e
[e
b e
b
b℄℄℄=[a[e
e
b℄℄=[ab[e
e
℄℄
=[ab℄
giving[(H;; 0
)℄=H,andompletingtheproof.
Notallnearheapshavespinesandthereforenotallnearheapsareoftheform
[S℄foraCliordsemigroupS. Anexampleofanearheapwithoutaspineisthe
free near heap generated by twoelements. A simpler example is the free fully
symmetrinearheapontwogenerators,whihwenowonstrut.
Example9. Thefreefullysymmetrinearheapontwogenerators.
Leta;b betwosymbolsandformtheniteset ofstrings
H(a;b)=fa;b;a 2
b;ab 2
g
(where a 2
denotes aa and so on), and dene a ternary operation on H(a;b) by
setting[w
1 w
2 w
3
℄tobetheresultofrstsortingintoalphabetialordertheletters
inthestringonatenationw
1 w
2 w
3
togivea m
b n
,wherem+nisneessarilyodd,
andthenreduingtoanelementofH(a;b)byreduingpowersmodulo2downto
1or2(orelse 0ifthatletterdidnotouratall).
Thusforexample[aaa℄=a,and
[(aab)(b)(abb)℄!a 3
b 4
!ab 2
:
Itiseasytoonrmthat H(a;b)isafullysymmetrisemiheapinthesense that
[w
1 w
2 w
3
℄=[w
1 w
2 w
3
℄
where
1
;
2
;
3
is anypermutation of 1;2;3. Moreoverit islearly idempotent,
andheneisanearheap.
We laim that H(a;b) has no spine. The ongruene has at least three
equivalenelassesonH(a;b),twoofwhiharefagandfbg. AnyspineforH(a;b)
mustthereforeinludebothaandb,although[aab℄=a 2
b6=ab 2
=[abb℄soaand
A similar though more ompliated argument an be used in the non-fully
symmetri ase, to establish that the free near heap on two generators is an
innitenearheapwithoutaspine.
Toompletelytnears heapsinto thepattern suggestedbyProposition1,we
mustonsider whether or notall nearheaps anbeembedded in anear heapof
theform[S℄forsomeCliordsemigroupS. Theanswertothatquestionis\yes"
asweshallprove.
4. Nearheaps as strong semilattiesofheaps
EveryCliordsemigroup S is a semilattie of groups,meaning that there is
aongruene on S for whih eah -lass is agroupand S= is asemilattie.
By Proposition 6, every near heap is a \semilattie of heaps" in the obvious
sense. But every Cliord semigroup is not only a semilattie of groups but a
strongsemilattieof groups asin[3℄;fullinformationaboutthemultipliationin
aCliordsemigroupS anbe obtainedfrom suh astrong semilattieof group
deomposition. ThusifS= S
e2E(S) S
e
isthedeompositionofS intogroupsS
e
(one for eah e 2 E(S)), then for every e;f 2 L with e f, there is a group
homomorphism
e;f :S
e
!S
f
forwhih
e;e
istheidentitymap onS
e ,and
foralle;f;g2Lforwhih ef g,
f;g Æ
e;f
=
e;g .
Onethen ndsthat fora
e 2S
e anda
f 2S
f ,a
e a
f
=
e;ef (a
e )
f;ef (a
f
)asalu-
latedin S
ef
, sothat informationaboutthemultipliationsineah ofthegroups
togetherwithallthehomomorphisms
e;f
ompletelydeterminesthemultiplia-
tiononS.
Oneandeneanabstratstrongsemilattieofgroupstobeanydisjointunion
ofgroupsS= S
e2L S
e
,Lasemilattie,equippedwithhomomorphismsasabove,
andwithmultipliationdened asfollows: foralla
e 2S
e anda
f 2S
f ,
a
e a
f :=
e;ef (a
e )
f;ef (a
f
)asalulatedinS
ef .
It then follows easily that S
e S
f S
ef
for all e;f 2 L. S anbe shown to be
asemigroup;indeed it isalwaysaCliord semigroup(witha 0
dened tobethe
uniqueb 2 S suh that aba =a;bab=b). Hene everyCliord semigroup is a
strongsemilattieof groups. Forthedetails,onsult[3℄.
Ofourse,inaCliordsemigroupS,thesemilattieL=E(S)isembeddedin
thesemigroup: itisbothasubsemigroupandaquotientsemigroup. IndeedLis
thespineofthenear heapS. However,aswehaveseen,notallnear heapshave
spines. Intheasesthatdo,therewillbesomesortofstrongsemilattieofheaps
representation. Theinterestisinthegeneralase.
A strong semilattie of heaps is dened to bea disjoint unionof heaps S =
S
e2L S
e
, where L is a semilattie, suh that there are heap homomorphisms
e;f :S
e
!S
f
foreahe;f 2Lforwhihef,andforwhih
e;e
istheidentitymap onS
e ,and
foralle;f;g2Lforwhih ef g, Æ = .
Suh anS is turned into a ternary algebrabysetting, forall a
e 2S
e , a
f 2S
f
anda
g 2S
g ,
[a
e a
f a
g
℄=[
e;efg (a
e )
f;efg (a
f )
g;efg (a
g )℄:
Notation: [S
e
;L;
e;f
℄.
Theorem10. Astrongsemilattieofheaps[S
e
;L;
e;f
℄isanearheapwhihis
asemilattieoftheheapsS
e
,withthesemilattieisomorphito L.
Proof: It is obvious that the S
e
are losed under the ternary operation on S
(andofourseareheaps). WenextshowS isasemiheap.
Leta
2S
foreah2fe;f;g;h;ig. Then
[[a
e a
f a
g
℄a
h a
i
℄
= [[
e;efg (a
e )
f;efg (a
f )
g;efg (a
g )℄a
h a
i
℄
= [
efg;efghi ([
e;efg (a
e )
f;efg (a
f )
g;efg (a
g )℄)
h;efghi (a
h )
i;efghi (a
i )℄
= [[
e;efghi (a
e )
f;efghi (a
f )
g;efg (a
g )℄
h;efghi (a
h )
i;efghi (a
i )℄
= [[
e;efghi (a
e )[
h;efghi (a
h )
g;efg (a
g )℄
f;efghi (a
f )℄
i;efghi (a
i )℄;
whih averysimilar routinealulationshowsis equalto[a
e [a
h a
g a
f
℄a
i
℄, and so
alsobysymmetryto [a
e a
f [a
g a
h a
i
℄℄, soS isasemiheap.
Weturntothenearheaplaws.Idempoteneisimmediate(sinethealulation
of[a
e a
e a
e
℄takesplae whollywithin S
e
, whihisaheap). Finally,
[a
e a
e a
f
℄
= [
e;ef (a
e )
e;ef (a
e )
f;ef (a
f )℄
= [
e;ef (a
e )
e;ef (a
e )
f;ef (a
f
)℄sinetheomputationisinside theheapS
ef
= [
e;ef (a
e )
f;ef (a
f )
f;ef (a
f
)℄againworkinginS
ef
= [a
e a
f a
f
℄
asrequired. It isobviousthat [S
e S
f S
g
℄S
efg
=S
[efg℄
foralle;f;g2L, sothe
partitionofS into thedisjointS
e
isaongruene,andthat S=
=
L.
Thisresultjustiestheterm\strongsemilattieofheaps". NotethatLinthis
proofisnotin generalrepresentedasasubsetof S,onlyasaquotient.
Thefollowing resultextends Theorem 4.2.1 of [3℄ stating that every Cliord
semigroupisasemilattieofgroups,toover\spineless"ases.
Theorem 11. Let H be aternary algebra, L a semilattie. The followingare
equivalent.
(1) H is anearheapwithL
= H=.
(2) H is asemilattieofheaps S
e2L H
e .
(3) H is astrongsemilattieofheaps[H ;L; ℄.
Proof: (1))(2)hasbeenshownalready.
For(2))(3), letH = S
e2L H
e
beasemilattieofheaps. Lete;f 2L, with
f e. Then for all a
e 2 S
e and a
f 2 S
f , [a
e a
f a
f
℄ 2 S
[eff℄
= S
f
. So dene
e;f : S
e
! S
f
by setting
e;f (a
e ) = [a
e a
f a
f
℄ for any a
f 2 S
f
. This is well-
dened (independent of the hoie of a
f 2 S
f
), beause if also b
f 2 S
f , then,
usingtheheaplawsasneeded,wehave
[a
e b
f b
f
℄ = [a
e [a
f a
f b
f
℄[a
f a
f b
f
℄℄
= [a
e [b
f a
f a
f
℄[b
f a
f a
f
℄℄
= [[a
e a
f a
f
℄b
f [b
f a
f a
f
℄℄
= [[[a
e a
f a
f
℄b
f b
f
℄a
f a
f
℄
= [[a
e a
f a
f
℄a
f a
f
℄sine[a
e a
f a
f
℄2S
f
= [a
e a
f a
f
℄:
Now
e;e
is theidentity onS
e
beauseforany a
e 2S
e ,
e;e (a
e )=[a
e b
e b
e
℄=a
e
foranyb
e 2S
e .
Wenext show
e;f
is ahomomorphismS
e
! S
f
. Soleta
e
;b
e
;
e 2S
e , with
d
f 2S
f
. ThenrepeatedlyusingtheheaplawsinS
f ,
e;f ([a
e b
e
e
℄) = [[a
e b
e
e
℄d
f d
f
℄
= [[[a
e b
e
e
℄d
f d
f
℄d
f d
f
℄ sineS
e S
f andS
f
isaheap
= [[a
e [d
f
e b
e
℄d
f
℄d
f d
f
℄
= [[a
e d
f d
f
℄[d
f
e b
e
℄d
f
℄
= [[[a
e d
f d
f
℄b
e
e
℄d
f d
f
℄
= [[a
e d
f d
f
℄b
e
e
℄:
However,
[
e;f (a
e )
e;f (b
e )
e;f (
e
)℄ = [[a
e d
f d
f
℄[b
e d
f d
f
℄[
e d
f d
f
℄℄
= [[[a
e d
f d
f
℄d
f d
f
℄b
e [
e d
f d
f
℄℄
= [[a
e d
f d
f
℄b
e [
e d
f d
f
℄℄
= [a
e [b
e d
f d
f
℄[
e d
f d
f
℄℄
= [[a
e [b
e d
f d
f
℄
e
℄d
f d
f
℄
= [[a
e d
f [d
f b
e
e
℄℄d
f d
f
℄
= [a
e d
f [[d
b b
e
e
℄d
f d
f
℄℄
= [a
e d
b [d
b b
e
e
℄℄
= [[a
e d
f d
f
℄b
e
e
℄
=
e;f ([a
e b
e
e
℄)
Finallywemustshowthatforalle;f;g2Lforwhihef g,
f;g Æ
e;f
=
e;g
. Sosupposee;f;g2Lsatisfyef g. Thenforanya
e 2S
e ,a
f 2S
f and
a
g 2S
g ,
(
f;g Æ
e;f )(a
e
) = [[a
e a
f a
f
℄a
g a
g
℄
= [[[a
e a
f a
f
℄a
g a
g
℄a
g a
g
℄
= [[[a
e [a
g a
f a
f
℄a
g
℄a
g a
g
℄
= [[a
e a
g a
g
℄[a
g a
f a
f
℄a
g
℄
= [[[a
e a
g a
g
℄a
f a
f
℄a
g a
g
℄
= [[a
e a
g a
g
℄a
f [a
f a
g a
g
℄℄
= [a
e [a
f a
g a
g
℄[a
f a
g a
g
℄℄
=
e;g (a
e )
sine[a
f a
g a
g
℄=
f;g (a
f )2S
g
. This ompletestheproofthat anysemilattieof
heapsis astrongsemilattieofheaps.
For (3) ) (1), the fat that H = [H
e
;L;
e;f
℄ is a near heap was shown in
Theorem 10. For eah a 2 H, let a
be the -lass ontaining a. To show
that L isthe sameasin Proposition6, itsuÆes to showthat theheapsH
e in
H =[H
e
;L;
e;f
℄are preiselythesubheapsa
ofH. ItsuÆesto showthatfor
alla2 H, ifa 2H
e then a
=H
e
. Sosuppose a2H
e
. Of ourseH
e a
by
maximalityof a
. Conversely, ifb 2a
, suppose b2 H
f
. Then[abb℄ =a, so in
partiular,H
e
3a=[aaa℄=[bba℄2H
ef
,soef =e, asotherwiseH
ef
\H
e
=;.
Bysymmetry(sinealsoa2b
)ef =f,soe=fandb2H
e
. Henea
H
e
.
Note that the homomorphisms
e;f
used to dene a given strongsemilattie
ofheapsH=[S
e
;L;
e;f
℄ (thatis,anearheapbytheaboveresult)areuniquely
determinedbythenearheap. First,themaximalsubheapdeomposition S
e2L H
e
(inluding L up to isomorphism) depends only on the struture of H, and for
a
e 2H
e anda
f 2H
f
,wehave[a
e a
f a
f
℄=[
e;ef (a
e )
f;ef (a
f )
f;ef (a
f )℄2H
ef , a
heap,andso[a
e a
f a
f
℄=
e;ef (a
e ),so
e;ef
iswhollydeterminedbythenearheap
operation. ThisparallelsthesituationforCliordsemigroups.
However,it follows from themain result of theprevioussetion that for any
near heap of the form [S℄ where S is aCliord semigroup, the struture of [S℄
ompletelydeterminestheCliord semigroupoperationsonS.
Corollary 12. Suppose S
1 and S
2
are two Cliord semigroups on the same
underlyingset forwhih[S
1
℄=[S
2
℄. ThenS
1
=S
2 .
Proof: First,itisaroutineexerisetohekthat,givenarepresentationofthe
CliordsemigroupS asastrongsemilattieofgroups,thereisaninduedrepre-
sentation of[S℄ asastrongsemilattieof heaps,using thesamesemilattie,the
subheaps assoiated with thesubgroups, and the samehomomorphisms. Then,
if S
1 and S
2
are twoCliord semigroupson the sameunderlying set for whih
[S
1
℄=[S
2
℄, thehomomorphismsinherited from S
1 and S
2
(as wellasthe S
e of
ourse)mustbethesame,andsoS andS arealsothesame.
Theorrespondingfatforarbitraryinvolutedsemigroupsfails: theinvoluted
semigroupoperationson S are notdetermined by the struture of [S℄. For ex-
ample,thezerosemiheaponaset,in whih allternaryprodutsarezero,arises
fromdistint,evennon-isomorphi,involutedsemigroupsontheset. Itwouldbe
interestingto determinethose varietiesV of involuted semigroupsforwhih the
operations on S 2 V are ompletely determined by [S℄ (at least up to isomor-
phism).
5. Embeddingnear heaps in Cliord semigroups
Aswehaveseen,Cliordsemigroupsgiverisetonearheaps,andindeedallof
the information presentin the Cliordsemigroupis retained bythe near heap.
However,noteverynearheapis[S℄whereS isaCliordsemigroup. Sowhatan
besaid? Canwegiveanembeddingtheoremfornearheaps,therebyprovidinga
ompletedentryin Proposition1?
Notethat the asesonsideredin Proposition1anall be dealtwith byrst
showingthateverysemiheapofagiventypemaybeembeddedinabi-unitalone
ofthesametype,andtheninvokingProposition2. Thisistheapproahtakenin
[5℄. However,thatapproahdoesnotreadilyextendtonearheaps.
First some observations about representationsin terms of partial mappings.
BytheWagner-Preston theorem, anyinversesemigroup Gis representable asa
subsemigroupofthesymmetrisemigroupofone-to-onepartialmapsX !X for
some set X. The atual representation used is a left regular one, whih maps
a2 Gto the partial map
a
: G! Ggivenby
a
(x)=ax forall x suh that
a 0
ax = x; when this is done,
a 0
a
is the restritionof the identity map to the
domainof
a and
aa 0
istherestritionoftheidentitymaptoitsrange.
RepresentingaCliordsemigroupinthisway,theinversesemigroupofpartial
mapshasthepropertythateverypartialmaphasequaldomainandrange(sine
aa 0
=a 0
a),andthat thepartialmapshavingagivendomainformagroup(sine
aa 0
=a 0
aisanidentityelement). Moreoverthebijetionsassoiatedwitha 0
aand
b 0
bagreeona 0
ab 0
b:thetwoheapsofmapsrestritdowntothesameheapofmaps
on thesmaller domain. This is aonrete wayto interpret the fat that every
Cliordsemigroupisasemilattieofheaps: thesemilattieisthesetofdomains
(=ranges)determinedbyE(G)=fa 0
aja2Gg,andtheheapsaretheassoiated
partialmapswithdomainsandrangesgivenbytheaa 0
.
Likewise, itis wellknown that everygeneralisedheap may be representedas
asemiheapofone-to-onepartial mapsX !Y (wherewithoutlossofgenerality
everyelementofxisin thedomainofoneofthemapsandeveryelementofy is
mappedtobyoneofthemaps): theoperationonsuhmapsis[fgh℄=fÆg 1
Æh.
Again, interpreting the near heap law shows that the maps an be organised
into subheapsaordingto their domains, andthose maps withagivendomain
also haveidential ranges(not equalto their domains this time, sinethey are
in dierent sets). For a xed represented near heap, let L
X
be the olletion
of domains and L the olletion of ranges: both sets are semilatties under
intersetion, as for generalised heaps in general. Again, it follows easily that
twosets of heaps (orresponding to twopossibledomains) restrit down to the
sameheapwhentheintersetionoftheirdomainsinL
X
isonsidered. Again,all
of this is nothing but a onrete realisationof Theorem 11: everynear heap is
asemilattieofheaps.
Wearenowinapositionto givethemain resultofthissetion.
Theorem 13. Everynear heap is embeddable in the semiheap obtained from
aCliordsemigroup.
Proof: Withoutlossofgenerality,letH beanearheapofpartialmapsX !Y
asdesribedabove. We shallshowhowto identifyX and Y in suhawaythat
theresultingCliordsemigroupembedstheoriginalnearheap.
Choosing S 2 L
X
, we have a xed set (indeed heap)of bijetions H
S from
S to S 0
2 L
Y
. Choose x 2 X and for any S 2 L
X
for whih x 2 S, dene
T
x
= fp(x) j p 2 H
S
g, a subset of Y independent of the hoie of S by the
restritionproperty. ThisanbeextendedtoarbitrarysubsetsofX intheobvious
way: forW X,deneT(W)= S
x2W T
x .
Likewise for y 2 Y, dene T 0
y
= fq(y) j q 1
2 H
S
g, where S X is suh
thaty 2f(S),and extendto subsetsofY asforT aboveto giveT 0
(S 0
). Nowif
a=q 1
(p(x))2T 0
(T
x
), thenq(a)=p(x)2T
x
,so T(T 0
(T
x ))T
x
,and beause
theoppositeinlusion obviouslyholds, wehaveT(T 0
(T
x ))=T
x
. It nowfollows
easily that thereis aone-to-one orrespondene betweensubsetsof theform T
x
inY andT 0
(T
x )in X.
Now suppose x 0
= 2 T
0
(T
x
). Suppose b 2 T 0
x
\T
x
. So b = p
1 (x
0
) = p
2 (x) for
some bijetions p
1 2 H
S
1
(where S
1 2 L
X
ontains x 0
) and p
2 2 H
S
2
(where
S
2 2 L
X
ontains x). Hene y = p 1
1 Æp
2
(x)2 T 0
(T
x
), aontradition. Hene
T
x
\T
x 0
=;. Similarlythen,S(T
x )\S(T
x 0
)=S(T
x
\T
x 0
)=S(;)=;. Thusthe
T
x
formapartitionofY andtheS(T
x
)formapartitionofX.
Notethat forany S2L
X
forwhih x2S,ifa2T 0
(T
x
),then a=q 1
(p(x))
for some p;q 2 H
S
, so a 2 S; heneT 0
(T
x
) S for everyS 2 L
X
ontaining
x. Pik p 2 H
S
and dene
x : T
0
(T
x ) ! T
x
by setting
x
(a) = p(a) for all
a 2 T 0
(T
x
), aone-to-one funtion (being a restrition of the bijetive funtion
p:S!f(S)). Itisalsosurjetive,asifb2T
x
,thena=p 1
(b)2T 0
(T
x
)satises
p(a)=b. (HeneonlyonehoieofbijetionwasreallyneededindeningT
x and
soon.)
Webuildabijetion :X !Y outofthebijetions
x
in theexpeted way:
(x)=
x
(x)forallx2X. Thisworks beausetheT 0
(T
x
)areapartitionof X
(andlikewisefortheT
x
in Y). Foronvenienewemakediretuseoftheinverse
bijetion= 1
,mappingY !X.
WenowmapH intotheinversesemigroupI(X)ofone-to-onepartialmappings
onX. Thuslet bethemappingtaking H into I(X)suh thatforeahf 2H,
(f) = Æf; learly (f) 2 I(X). We show is an embedding of H into the
Forf;g;h2H,
[(f)(g)(h)℄ = (f)Æ(g) 1
Æ(h)
= (f)Æ(Æg) 1
ÆÆh
= (f)Æg 1
Æ 1
ÆÆh
= ÆfÆg 1
Æh
= Æ[fgh℄
= ([fgh℄):
Soisahomomorphismwhihisobviouslyinjetive(sine isabijetion).
Now letM be the inverse subsemigroup of I(X)generated by H
1
= f(f) j
f 2Hgunder theoperationsofinversionandomposition.
Note that eah (f) 2 M (where f 2 H) has equal domain and range, so
(f)Æ(f) 1
= (f) 1
Æ(f), and if also g 2 H, then (f)Æ(f) 1
Æ(g) =
([ffg℄)=([gff℄)=(g)Æ(f) 1
Æ(f). AtypialelementofMisaomposite
w=a
1 a
2 a
n
ofelementsofI(X)of theform (f)or(f) 1
forsomef 2H,
and for suh elements we havejust shown that xx 0
=x 0
x and xy 0
y = yy 0
x. It
therefore follows easily that ww 1
= (a
1 a
2 a
n )(a
1
n a
1
2 a
1
1
) whih easily
rearrangesto(a
1 a
1
1 )(a
2 a
1
2
)(a
n a
1
n
),whih bysymmetryalsoequalsw 1
w.
HeneM isaCliordsemigroup,embeddingH.
Referenes
[1℄ BaerR.,ZurEinf uhrungdesSharbegris,J.ReineAngew.Math.160(1929),199{207.
[2℄ HawthornI.,StokesT.,Radialdeompositionsofsemiheaps,Comment.Math.Univ.Car-
olin.50(2009),191{208.
[3℄ HowieJ.M.,FundamentalsofSemigroupTheory,OxfordUniversityPress,Oxford,1995.
[4℄ PruferH.,TheoriederAbelshenGruppen,Math.Z.20(1924),165{187.
[5℄ Wagner V.V., The theory of generalized heaps and generalized groups (Russian), Mat.
SbornikN.S.32(1953),545{632.
[6℄ Wagner V.V., On the algebrai theory of oordinate atlases, II (Russian), TrudySem.
Vektor.Tenzor.Anal.14(1968),229{281.
DepartmentofMathematis, TheUniversityofWaikato,Private Bag3105,
Hamilton,NewZealand
E-mail: stokesmath.waikato.a.nz
(Reeived September15,2010 , revised February2,2011 )
