Partial order embeddings with convex range

0.63 MB·English

by  James Hirschorn

PARTIAL ORDER EMBEDDINGS WITH CONVEX RANGE
JAMES HIRSCHORN
7
0
Abstra
t. A
arefulstudyismadeofembeddingsofposetswhi
hhavea
onvex
0
range. We observe that su
h embeddings share ni
e properties with the homo-
2
morphismsof morerestri
tive
ategories; for example,we show thateveryorder
n
embeddingbetweentwo latti
es with
onvexrange isa
ontinuous latti
e homo-
a
J morphism. Anumberof pσos:etNsNar→e
NoNnsidered; forexample,we provethatevery
produ
t order embedding with
onvexrange is of the form
7 σ(x)(n)=`(x◦gσ)+yσ´(n) n∈Kσ,
1 (0.1) if
σ(x)(n) = yσ(n) x ∈ NN Kσ ⊆ N gσ : Kσ → N
] and yσ ∈NoNtherwise, for all , where , is
A a bije
tion and . Themost
omplexposetexaminedhereisthequNoItient
of the latti
e of Baire measurable fun
tions, with
odomain of the form for
R I
some indexset , modulo equality on a
omeager subset of thedomain, with its
.
h `natural' ordering.
t
a
m
[

Contents
1
v
1. Overview 2
6
8 1.1. Terminology 4
4 2. The quasi ordering of a monoid 6
1
2.1. Distributive laws 9
0
3. Order embeddings with preregular range 12
7
0 3.1. Preregularity 12
/ 3.2. Order re(cid:29)e
ting homomorphisms 14
h
t 3.3. Continuity 15
a
m 3.4. Bases of latti
es 19
3.5. Extensions of embeddings 22
:
v 4. The partial orders 28
i
X 4.1. Power set algebras 28
N
r 4.2. Powers of 29
a
4.3. Category algebras 31
N
4.4. Baire fun
tions into powers of modulo almost always equality 37
N
4.5. Continuous fun
tions into powers of 50
Date: January2, 2007.
2000 Mathemati
s Subje
t Classi(cid:28)
ation. Primary 06A06; Se
ondary 03E15, 03E40, 06A11,
06B30, 06F05, 54C35.
Key words and phrases. Partial order, embedding,
onvex, S
ott topology, irrationals, Baire
measurable,
omplete semilatti
e, monoid.
The author a
knowledges the generous support of the Japanese So
iety for the Promotion of
S
ien
e (JSPS Fellowship for Foreign Resear
hers, ID# P04301).
1
2 JAMESHIRSCHORN
4.6. Further dire
tions 53
Referen
es 56
1. Overview
The abstra
t obje
ts of study here are embeddings between arbitrary posets hav-
ing preregular (e.g.
onvex, see de(cid:28)nition 3.11) range. The main observation is that
they are
ontinuous with respe
t to a natural poset topology (namely the S
ott
topology). This observation is then applied to investigate embeddings with
onvex
range between some spe
i(cid:28)

lasses of posets.
It turns out that the determination of these embeddings for various examples
of posets generalizes known results about homomorphisms in more restri
tive
ate-
σ :P(X) → P(Y)
gories. For example, we show in Ÿ4.1 that every order embedding
σ(a) = h[a] ∪ b h : X → Y
with
onvex range is of the form for some inje
tion
b ⊆ Y
and some ; this
an be
ompared with the known fa
t that every
ontinuous
σ : P(X) → P(Y) σ(a) = h−1[a]
Boolean algebra monomorphism is of the form for
h :Y 99K X
some partial surje
tion .
By de(cid:28)nition, an order embedding is an order preserving map (i.e. partial order
homomorphism) that is an isomorphism onto its range. Thus for order theoreti
stru
tures, e.g. latti
es and Boolean algebras, where the stru
ture is
ompletely
determined by the ordering, an order embedding is in fa
t an isomorphism for the
given stru
ture onto its range. Hen
e
hara
terizing these embeddings involves two
aspe
ts: determining the isomorphisms of the given stru
ture; and determining the
range of these embeddings. For example, it is a spe
ial
ase of the above mentioned
P(X) P(Y)
known fa
t, that Boolean algebra isomorphisms between and are of the
σ(a) = h[a] h : X → Y
form for some bije
tion . Thus the `new result' in the above
σ :P(X) →

hara
terization
on
ernstherange,namelythateveryorderembedding
P(Y) [σ(∅),σ(X)] = {a ⊆ Y :
with
onvex range has range equal to the interval
σ(∅) ⊆ a⊆ σ(X)}
. (Of
ourse, the
omputation of the range in thisexample follows
trivially from the de(cid:28)nition of
onvexity, but in slightly more
omplex examples
this
omputation
an be
ome di(cid:30)
ult). However, this is an in
omplete view of
the situation where there is additional stru
ture that is not purely order theoreti
.
Indeed many our examples are also monoids.
N {0,1,...}
Write for the set of nonnegative integers. The usual (linear) partial
N 0 < 1 < ···
ordering of is . The example most imNportant to us is the
lass of
N
partial order embeddings between the irrationals , i.e. the set of all fun
tions
N N
from into (see e.g. [Hir06a℄), with the produ
t order. Indeed this paper is
the se
ond part of a series, where the third part [Hir06a℄ is entitled (cid:16)Chara
terizing
the quasi ordering of the irrationals by eventual dominan
e(cid:17) (and it is the sequel
to [Hir06b℄), where order embeddings of the irrationals with
onvex range play a

ru
ial role in
hara
terizing the eventual dominan
e ordering (
f. Ÿ4.6.4) in terms
of the produ
t order.
PARTIAL ORDER EMBEDDINGS WITH CONVEX RANGE 3
NI NJ
More generally, we
onsider embeddings between the produ
t orders and
I J x,y ∈ NI
for arbitrary index sets and . Thus for all ,
x≤ y iff x(i) ≤ y(i) i∈ I.
(1.1) for all
NI NJ
In this
ase we as
ertained (
f. Ÿ4.2) that every order embedding from into
with
onvex range is in fa
t a monoid embedding for
oordinatewise addition plus a
NJ

onstant in .
Further generalization was desired to fun
tion spa
es with the irrationals as the
X Y Baire(X,Y)

odomain. Let and betopologi
alspa
es. Welet denotethefamily
X Y
of all BaireNmeasurableNfun
tions from into . In Ÿ4.4 we
onsider quotients of
Baire(X,N ) N
(where has the produ
t topology) over the equivalen
e relation of
=
aa
(cid:16)almost always(cid:17) equality :
f = g f(z) = g(z) z ∈X
aa
(1.2) if for almost all ,
N
z ∈ X Baire(X,N )/=aa
or in other words for
omeagerly many ; where is given
≤ ≤
aa
the order indu
ed by :
[f]≤ [g] iff f(z)≤ g(z) z ∈ X.
aa
(1.3) for almost all
N N
Baire(X,N )/=aa Baire(Y,N )/=aa
The embeddings of into with
onvex range are
des
ribed pre
isely, where, as far as we know, even the isomorphism stru
ture was
not previously known. This involves
hara
terizing embeddings with
onvex range
of
ategory algebras of topologi
al spa
es, or equivalently the regular open algebras
of these spa
es, in Ÿ4.3. We note that there are some subtleties in generalizing to
Baire(X,NI)/=aa
arbitrary index sets, i.e. to , that are addressed there.
Our interest in quotients of Baire fun
tions stems from set theoreti
for
ing. For
I
example, for an index set in the `referen
e model'(cid:22)
alled the ground model in the
NI
terminology of set theoreti
for
ing, it is well known that every member of in the
extension of this model obtained by Cohen for
ing, is determined by a member of
Baire(R,NI)/=aa
in the ground model. Whilst in the other dire
tion, we shall use
(Baire(X,NI)/=aa,≤aa)
settheoreti
for
ing toprove that isa
omplete semilatti
e
NI
satisfyingmanyofthesamepropertiesas (theorem4.40). This
anbegeneralized
S P(S)
extensively; for example, for in the ground model, every member of in the
Baire(R,P(S))/=aa
Cohen model
orresponds to a member of (
f. Ÿ4.6.1), and by
repla
ing (cid:16)almost always(cid:17) with (cid:16)almost everywhere(cid:17) in the measure theoreti
sense,
we obtain a
orresponden
e with random for
ing (
f. Ÿ4.6.2).
In Ÿ2 we study a standard asso
iation of a quasi order to every monoid. We
are espe
ially interested in those monoids(cid:22)we
all them latti
e monoids(cid:22)where
the asso
iated quasi order is in fa
t a latti
e. Then various in(cid:28)nite distributive
laws are examined for these latti
e monoids. The main purpose of this se
tion for
the present paper, is that it allows us in Ÿ3.5 to use algebrai
methods to extend
embeddings from a suitably dense subset of some latti
e monoid to the entire latti
e
(
f.
orollary 3.98). This in turn is applied in Ÿ4.5 to use our des
ription in Ÿ4.4 of
Baire(X,NI)/=aa Baire(Y,NJ)/=aa
the embeddings of into with
onvex range in
order to obtain a pre
ise des
ription of the
lass of embeddings with
onvex range
C(X,NI) C(Y,NJ)
of into . That is, the family of
ontinuous fun
tions ordered
4 JAMESHIRSCHORN
pointwise. We also see examples (e.g. lemma 3.16, theorem 3.59) of how algebrai
properties of subsets of latti
es may entail order theoreti
regularity properties.
The general theory is in Ÿ3, in
luding the S
ott
ontinuity of embeddings with
preregularrange,andnumerous
onsequen
esofthisresult. InŸ3.4,westudyvarious
notionsofdensenessinalatti
e,andthisisappliedinŸ3.5toextendhomomorphisms
from suitably densesubsetsof alatti
ewhile preserving variousdesirableproperties.
(P(N)/Fin,⊆∗)
Further d(NireN
/tiFonins,f≤or∗)the Boolean algebra and its
lose relative
the latti
e are suggested in Ÿ4.6.3 and Ÿ4.6.4.
1.1. Terminology. A homomorphism refers to an arrow (i.e. morphism) of the
intended
ategory, while an isomorphism is an invertible homomorphism. For a
(C,U) h : c → d

on
rete
ategory , we say that an embedding is a homomorphism
g c
for whi
h there exists some invertible homomorphism with domain su
h that
Uh(x) = Ug(x) x ∈ Uc
for all . Thus in a
ategory where the homomorphisms are
f : X → Y
fun
tions, an embedding is a homomorphism that is an isomorphism
ran(f) f′ : X → ran(f)
onto its range, i.e. is an obje
t of the
ategory and given
f′(x) = f(x)
by is an isomorphism. Monomorphisms (i.e. moni
s) refer to homo-
f
morphisms that are left
an
ellative under
omposition, i.e. is a monomorphism
f ◦g = f ◦h g = h g,h f
i(cid:27) implies for all with
odomain equal to the domain of .
And epimorphisms (i.e. epis) refer to homomorphisms that are right
an
ellative.
In all of the
on
rete
ategories
onsidered below, monomorphisms are simply the
inje
tive homomorphisms, and in all of these
ategories with the ex
eption of the
monoids, epimorphisms are pre
isely the surje
tive homomorphisms.
(O,≤) ≤
A quasi order (also often
alled a preorder) is a pair where is a re(cid:29)exive
O (O,≤) <
and transitive relation on . For a quasi order , we write for the relation
p < q p ≤ q q (cid:2) p
de(cid:28)ned by if and . Note this disagrees with another usage where
p < q p ≤ q p 6=q
i(cid:27) and . A poset (partial order) is a quasi order where the relation
p ≤ q q ≤ p p = q (P,≤) p < q
is also antisymmetri
(i.e. and imply ). In a poset ,
p ≤ q p 6= q
i(cid:27) and . A poset with a minimum element is
alled a pointed poset.
(O,≤) A ⊆ O (A,≤)
Note that if is a quasi order (poset) then for every subset , is
also a quasi order (poset). We
all it a quasi suborder (subposet). In the
ategory
of quasi orders, the homomorphisms are order preserving maps, i.e. for two quasi
(O,≤) (Q,.) σ :O → Q p ≤ q σ(p). σ(q)
orders and , isorder preserving if implies
p,q ∈ O
for all . Thus isomorphisms are bije
tions that are both order preserving
σ : O → Q σ(p) . σ(q)
and order re(cid:29)e
ting, where is order re(cid:29)e
ting if implies
p ≤ q p,q ∈ O A ⊆ O
for all . A subset is
alled bounded above, or just bounded, if it
p ∈O a≤ p a∈ A
has an upper bound, i.e. some su
h that for all . And we say that
A q ∈ O q ≤ a
is bounded below if it has a lower bound, i.e. some su
h that for all
a ∈ A
.
The
lass of posets is viewed as a full sub
ategory of the quasi orders. Noti
e
(P,≤) (Q,.) σ : P → Q
that for a poset and a quasi order, is an embedding i(cid:27) it is
A
both order preserving and re(cid:29)e
ting. Wewrite for the supremum, i.e. minimum
A ⊆ P A
upper bound, of a subset (whi
h may oWr may not exist), and we write
a∨b a∧b {a,b}
for the in(cid:28)mum, i.e. maximum lower bound. We write and for V
W
PARTIAL ORDER EMBEDDINGS WITH CONVEX RANGE 5
{a,b} σ : P → Q
and , respe
tively. A fun
tion between two posets is
alled join-
σ(p ∨q) = σ(p)∨σ(q) p∨q p,q ∈ P
preseVrving if whenever exists, for all (i.e. if
p∨q σ(p)∨σ(q)
exists then so does satisfying the equation). The dual notion is
meet-preserving.
We take a join semilatti
e, whi
h we also just
all a semilatti
e, to be a poset
(L,≤) ∨ L p∨q p,q ∈ L
su
h that is a binary operation on , i.e. exists for all . And a
meet semilatti
e is de(cid:28)ned dually. A latti
e is a poset that is both a semilatti
e and
(L,≤) (A,≤)
a meet semilatti
e. A subsemilatti
e of a semilatti
e is a semilatti
e ,
A⊆ L (A,≤) |= pc =a∨bq (L,≤)|= pc = a∨bq
where , su
h that i(cid:27) , i.e. the supre-
(A,≤) L
mum
omputed in the order agrees with the supremum taken in . Meet
A ⊆ L
subsemilatti
es and sublatti
es are de(cid:28)ned analogously. Note that a subset
(L,≤)
may be a latti
e as a subposet of , without being a sublatti
e. The homomor-
phisms of the
ategory of semilatti
es are the join-preserving fun
tions, while the
homomorphisms of the
ategory of meet semilatti
es are the meet-preserving fun
-
p∨q
tions. Notethatthesearebothsub
ategoriesoftheposet
ategory, be
ause and
p∧q p ≤ q q ≤ p q 6= p
do not exist when , and , and be
ause join/meet semilatti
e
p ≤ q p∨q = q p∧q = p
homomorphismsareorderpreserving, as i(cid:27) i(cid:27) . The
ategory
of latti
es is the interse
tion of the
ategories of join and meet semilatti
es, i.e. the
homomorphisms are the fun
tions that are both join and meet-preserving. It is easy
to (cid:28)nd a
ounterexample showing that a quasi order homomorphism between two
latti
es need not be a latti
e homomorphism (i.e. latti
es are not a full sub
ategory
of the quasi orders). On the other hand, sin
e the latti
e operations are obviously
determined by the ordering of the latti
e, a quasi order isomorphism between two
latti
es (equivalently, a poset isomorphism) is in fa
t a latti
e isomorphism. Take
note that latti
e embeddings are the same thing as latti
e monomorphisms (and
similarly for join/meet semilatti
e embeddings), i.e. they are the inje
tive latti
e
homomorphisms (this is not true of the
ategory of posets). Moreover:
σ
Proposition 1.1. A latti
e homomorphism is an embedding i(cid:27) it is stri
tly order
p < q σ(p) < σ(q)
preserving, i.e. implies .
(L,≤) A
By a
omplete semilatti
e we mean a join semilatti
e su
h that exists
A ⊆ L
whenever is bounded. Note that a
omplete semilatti
e is poinWted, with
∅ 0
minimumelement ,whi
hwedenoteby . Noti
ealsothata
ompletesemilatti
e
A A 6= ∅
is in fa
t a latti
e,Wand moreover exists whenever (see e.g. [DP02℄). A
A A A⊆ L

omplete latti
e is a latti
e su
h thVat and exist for every subset . By
adding a top element to any
ompleteWsemilattVi
e one obtains a
omplete latti
e. A
latti
e is
alled bounded if ithas both amaximum and minimum element; we denote
1
the maximum element by .
(B,≤) p ∈ B
We take a Boolean algebra to be a bounded latti
e su
h that every
−p p∧−p = 0 p∨−p = 1
has a
omplement, whi
h we write as , satisfying and .
Homomorphismsinthe
ategoryofBooleanalgebrasarelatti
ehomomorphismsthat
0 1
preserve
omplements (and thus also preserve and ). Sin
e the Boolean algebra
operationsare
ompletelydeterminedbytheorder,everyposetisomorphismbetween
two Boolean algebras is in fa
t a Boolean algebra isomorphism. In the
ategory of
Boolean algebras embeddings and monomorphisms
oin
ide.
6 JAMESHIRSCHORN
(M,·)
Re
allthatamonoid isasemigroupthathasanidentity. Wedonotneedto
e
spe
ifytheidentitybe
auseitisuniquelydetermined; wedenoteitby . Amonoidis
a·b = a·c b = c b·a =c·a b = c a,b,c ∈ M

an
ellative if implies and implies ,forall .
a∈ M a−1 a·b−1
The inverse of , when itexists, isdenoted ; more generally, we write
c ∈ M a = c·b
for the element su
h that , if it exists. It is uniquely determined
M (M,+)
so long as is
an
ellative. We write when dealing with a
ommutative
− 0
monoid; use to denote the inverse; and we use to denote the identity. A monoid
homomorphismismappreservingboththemonoidoperationandtheidentity. Inthe

ategory of monoids embeddings and monomorphisms both
oin
ide with inje
tive
S
homomorphisms. When we say that a subset of a
ommutative monoid is
losed
a−b ∈ S a,b ∈ S a−b
under subtra
tion we of
ourse mean that whenever and
exists. Sin
e every
an
ellative
ommutative monoid embeds into an Abelian group
(lemma 2.13), a submonoid of a
an
ellative
ommutative monoid that is
losed
under subtra
tion
an be viewed as a `subgroup'.
(G,+) M ⊆ G
Proposition 1.2. Let be an Abelian group, and be a submonoid.
S ⊆ M hSi∩M = S hSi
Then a subsemigroup is
losed under subtra
tion i(cid:27) , where
S
denotes the subgroup generated by .
S hSi = S−S
Proof. Sin
e is a subsemigroup of an Abelian group, . It immediately
hSi∩M = S S (cid:3)
follows that i(cid:27) is
losed under subtra
tion.
R X n
Re
all that a binary relation on a set is a
ongruen
e on some -ary
S X S(x ,...,x ) S(x′,...,x′ ) x R x′
relation on , if 0 n−1 i(cid:27) 0 n−1 , whenever k k for
k = 0,...,n − 1 n f X
all ; and it is a
ongruen
e on some -ary fun
tion on , if
f(x ,...,x ) R f(x′,...,x′ ) x R x′ k = 0,...,n − 1
0 k−1 0 k−1 whenever k k for all . A
∼ (O,≤) ≤

ongruen
e on some quasi order , i.e. a
ongruen
e on , that is moreover
O (O,≤)
an equivalen
e relation on is an orderable partition of in the terminology
≤
of [DR81℄, and it is
alled so be
ause determines a well de(cid:28)ned ordering of the
O /∼ [p] ≤ [q] p ≤ q (L,≤)
quotient via if . A latti
e
ongruen
e on some latti
e
L
means a relation on that is a
ongruen
e for both of the binary latti
e operations.
∼
Alatti
e
ongruen
e thatisalsoanequivalen
erelationindu
esalatti
estru
ture
L/∼ [a]∧[b] = [a∧b] [a]∨[b] = [a∨b]
on , with and . And a Boolean algebra
on-
gruen
e is a latti
e
ongruen
e that is also a
ongruen
e for the unary
omplement
operator. Of
ourse a Boolean algebra
ongruen
e that is an equivalen
e relation on
(B,≤) B/∼ −[a]= [−a]
determines a quotient Boolean algebra with .
(M,·)
Similarly, amonoid
ongruen
e foramonoid isa
ongruen
eonthemonoid
· ∼ ·
operation , and an equivalen
e relation that is a
ongruen
e on determines a
(M /∼,·)
quotient monoid .
2. The quasi ordering of a monoid
All monoids have a naturally asso
iated quasi ordering. This is well known,
e.g. [Kan69℄ to give one example. In this se
tion we examine some basi
properties
of this asso
iated quasi ordering, and introdu
e the notion of a latti
e monoid.
Then we fo
uson distributivity, where we prove that a large
lass of monoids satisfy

ertain in(cid:28)nitary distributive laws; for example, in
orollary 2.30 we prove that in
PARTIAL ORDER EMBEDDINGS WITH CONVEX RANGE 7
every
an
ellative
ommutativemonoid,additiondistributesoverarbitrarysuprema.
AsfarastheexamplesofŸ4are
on
erned, theresultsofthisse
tionareonlyapplied
to the latti
e monoid of Ÿ4.5. However, latti
e monoids will play a bigger role in the
sequel to this paper.
(M,·) ≤ M
(M,·)
De(cid:28)nition 2.1. A monoid hasan asso
iated quasi order on de(cid:28)ned
by
x≤ y x·a= y a∈ M
(M,·)
(2.1) if for some .
(M,·)
Proposition 2.2. If is a monoid the the relation de(cid:28)ned in equation (2.1) is
a quasi ordering.
(M,·) (M,·)
Proof. Itisre(cid:29)exivebe
ause hasanidentity, anditistransitivebe
ause
(cid:3)
is a semigroup.
Proposition 2.3. For any monoid, the identity is a minimum element of the asso-

iated quasi order.
a∈ M e·a= a e≤ a (cid:3)
(M,·)
Proof. For all , implies .
The quasi order asso
iated with a
an
ellative monoid has no maximal elements
unlessthemonoidisagroup,inwhi
h
asetheasso
iatedquasiorderisthe
omplete
quasi order.
(M,·) (M,≤ )
(M,·)
Lemma 2.4. Let be a
an
ellative monoid. If has a maximal
(M,·)
element then is a group.
a ∈ M b ∈ M a ≤ a ·b
(M,·)
Proof. Let be a maximal element. Take any . Then
a · b ≤ a a · b · c = a c ∈ M b · c = e
(M,·)
implies and thus for some . Hen
e by
(M,·)

an
ellativity. The existen
e of right inverses for all elements entails that is
(cid:3)
a group.
(M,·) b ≤ c a·b ≤ a·c
(M,·) (M,·)
Proposition 2.5. Let be a monoid. Then implies ,
a,b,c ∈ M
for all .
b ≤ c d b ·d = c (a ·b)· d =
(M,·)
Proof. Assuming , there exists su
h that . Thus
a·(b·d) = a·c (cid:3)
.
We are mostly interested in
ommutative monoids so that we at least have mono-
toni
ity. Indeed, the
ategory of positively quasi ordered
ommutative monoids is
widely studied in Ordered Algebra, see e.g. [Weh92℄ where they are named POM's,
≤
(M,+)
and the quasi order asso
iated with a
ommutative monoid is the minimal
(M,+,≤ )
(M,+)
quasi ordering su
h that is a POM.
(M,+) a ≤ b
(M,+)
Proposition 2.6. Let be a
ommutative monoid. Then implies
a+c≤ b+c a,b,c ∈ M
(M,+)
, for all .
a+d= b (a+c)+d =a+d+c = b+c (cid:3)
Proof. Write . Then .
(M,·)
De(cid:28)nition 2.7. A poset monoid is a monoid whose asso
iated quasi order is
in fa
t a partial order.
8 JAMESHIRSCHORN
e
Proposition 2.8. Every poset monoid is pointed with the minimum element.
(cid:3)
Proof. Proposition 2.3.
Lemma 2.9. Poset monoids do not have invertible elements besides the identity.
a ∈ M a·a−1 = e a ≤ e
(M,·)
Proof. Suppose that is invertible. Then implies , and
e ≤ a a= e (cid:3)
(M,·)
by proposition 2.3. Thus by antisymmetry.
Corollary 2.10. Can
ellative poset monoids do not have maximal elements, with
{e}
the ex
eption of the singleton monoid .
(cid:3)
Proof. Lemmas 2.4 and 2.9.
(M,·)
De(cid:28)nition 2.11. A monoid is
alled a semilatti
e monoid if the asso
iated
quasiorderisa
tually asemilatti
e, anditis
alledalatti
e monoid iftheasso
iated
quasi order is moreover a latti
e.
0= e
Remark 2.12. Note that in a semilatti
e monoid by proposition 2.8. Thus our
0
usage of as the additive identity in a
ommutative monoid is
onsistent with its
usage as the minimum element of a pointed semilatti
e.
We re
all the following basi
fa
t and provide a proof.
Lemma 2.13. A
ommutative monoidis embeddable (as amonoid) in some Abelian
group i(cid:27) it is
an
ellative.
(M,+) M−
Proof. Let be
ommutativeand
an
ellative. Let denotethesubgroupof
M N = (M\M−)∪{0}
invertible elements of . Notethat
ommutativity entailsthat
(M,+) M ×N
is a submonoid of . Now
onsider the produ
t monoid modulo the
relation de(cid:28)ned by
(a,b) ∼ (a¯,¯b) ∃r,r¯∈ N a+r = a¯+r¯ and b+r =¯b+r¯.
(2.2) if
∼ 0 ∈ N
Note that is an equivalen
e relation: im(ap,bli)es∼r(ea¯(cid:29),e¯bx)ivity,(a¯sy,¯bm)m∼et(ra¯¯y,¯¯bi)s
obvious, andwehavetransitivitybe
ausesupposing and ,
a+r = a¯+r¯ a¯+s= a¯¯+s¯ a+r+s= a¯+r¯+s=
a¯¯+r¯+s¯ and b+r+whsi=
h¯¯bw+itrh¯+
os¯mmutativity imply ∼
, and similarly . And using
ommutativity similarly,
(M ×N)/∼
is a
ongruen
e for addition (
f. Ÿ1.1). Therefore is a quotient monoid.
Observe that
[b,b] = [0,0] b ∈N
(2.3) for all ,
[0,0] [(0,0)]
and of
ourse , i.e. , is the identity.
((M ×N)/∼,+)
Claim 2.14. is an Abelian group.
Proof. It is
ommutative be
ause it is the quotient of a
ommutative monoid. Take
[a,b] −[a,b]
in the quotient. We prove that it has an inverse . First suppose that
a ∈ M− −[a,b] = [−a + b,0] [a,b] + [−a + b,0] = [b,b] = [0,0]
. Then be
ause
−[a,b] = [b,a] [a,b]+[b,a] = [a+b,a+b] (cid:3)
by (2.3). Otherwise be
ause .
a7→ [a,0] M (M ×N)/∼
Claim 2.15. is a monomorphism between and .
PARTIAL ORDER EMBEDDINGS WITH CONVEX RANGE 9
a+ b [a+b,0] = [a,0] + [b,0]
Proof. is mapped to and hen
e it a monoid homo-
[a,0] = [b,0] r s N
morphism. Now suppose . Then there are and in su
h that
a + r = b + s 0 + r = 0 + s r = s
, and , i.e. . And by the
an
ellative property
a+r = b+r a= b (cid:3)
implies as needed.
Claims 2.14 and 2.15 establish one dire
tion of the lemma, and it is
lear that a
(cid:3)
non
an
ellative monoid
annot be embedded in a group.
This is used to show that
an
ellative
ommutative monoids satisfy monotoni
ity
for subtra
tion.
(M,+)
Proposition 2.16. Let be a
an
ellative
ommutative monoid. Whenever
a−c b ≤ c a−b a−b ≥ a−c a,b,c ∈M
(M,+) (M,+)
exists, implies exists and , for all .
M
Proof. By lemma 2.13, we may assume that is a submonoid of some Abelian
(G,+) b ≤ c b + d = c d ∈ M
(M,+)
group . Now implies for some , and thus
(a − c) + d = (a − (b + d)) + d = a − b
as wanted, by the basi
properties of a
(cid:3)
group.
(M,+)
Proposition 2.17. Let be a
an
ellative
ommutative monoid. Whenever
a−c a ≤ b b−c a−c≤ b−c a,b,c ∈M
(M,+) (M,+)
exists, implies exists and , for all .
a+d = b
Proof. Write . Then sin
e by lemma 2.13 we
an assume we are working
(a−c)+d= (a+d)−c = b−c
inside an Abelian group, by basi
properties of an
(cid:3)
Abelian group.
(L,∨,∧) a∧(b∨c)=
2.1. Distributive laws. Re
all that a latti
e isdistributive if
(a∧b)∨(a∧c) a,b,c ∈ L
for all , and that:
(L,∨,∧) a∨(b∧c) = (a∨b)∧(a∨c)
Proposition 2.18. is distributive i(cid:27) for all
a,b,c ∈ L
.
(M,·) · ∨
De(cid:28)nition 2.19. Wesay that amonoid isleft -distributive over , or simply
(·,∨)
left -distributive, if
a·(b∨c) = (a·b)∨(a·c) b∨c
(2.4) whenever exists
a,b,c ∈ M · ∨
for all , and it is right -distributive over if
(a∨b)·c = (a·c)∨(b·c) a∨b
(2.5) whenever exists
a,b,c ∈ M · ∨ (·,∨)
for all , while it is -distributive over (or just -distributive) if it is
· ∨ · ∧
both left and right -distributive over . The de(cid:28)nition of -distributivity over is
exa
tly analogous.
(·,∨∧) (·,∨) (·,∧)
Let us say that it is -distributive if it is both -distributive and -
(·,∨∧)
distributive. We will say that a latti
e monoid is distributive if it is both -
distributive and distributive as a latti
e.
· ∨
Note that while we are mostly interested in -distributivity over in semilatti
e
· ∧
monoids, and -distributivity over is meet semilatti
e monoids, the de(cid:28)nition does
make sense for arbitrary monoids.
(+,∨) (+,∨)
Of
ourse, the notions of left -distributivity, right -distributivity and
(+,∨) (M,+)
-distributivity all
oin
ide for any
ommutative monoid , and similarly
+ ∧
for -distributivity over .
10 JAMESHIRSCHORN
(M,+) (+,∧)
Lemma 2.20. Let be a
ommutative -distributive monoid. Then
a∧b = 0 a∨b = a+b a,b ∈ M
implies , for all .
a,b ≤ a+b = b + a a∧ b = 0 d ≥ a,b
Proof. By
ommutativity, . Suppose and .
c a + c = d (+,∧) c = (a ∧ b) + c =
Let satisfy . Then by right -distributivity,
(a+c)∧(b+c) = d∧(b+c) ≥ b d≥ a+b (cid:3)
implies by proposition 2.5.
(M,+) +
Lemma 2.21. Let be a
ommutative
an
ellative -distributive monoid
∨ ∧ a∧c= 0 b∧c = 0 (a+b)∧c = 0
over and . Then and imply .
a∧c = b∧c = 0 d ≤ (a+b),c
Proof. Suppose that , and take . By lemma 2.20 and
+ ∨ a+b+d = a+(b∨d) = (a+b)∨(a+d)
-distributivity over , , and similarly
a+b+d= (a+b)∨(b+d)
. Thus
a+b+d =(a+b)∨(a+d)∨(b+d).
(2.6)
a+b +2d = (a+ d)+(b + d) = (a∨d)+ (b ∨d) = (a+ b)∨
But we also have
(a+d)∨(d+b)∨(d+d) ∨ 2d ≤ a+b+d
by distributivity over . Therefore, as sin
e
d ≤ a+b a+b+2d = a+b+d d= 0 (cid:3)
, we obtain , and thus by
an
ellativity.
2.1.1. In(cid:28)nite distributive laws.
a∧B =
Notation 2.22. We extend binary operations to sets in the usual way, e.g.
{a∧b : b ∈ B} A·b = {a·b :a∈ A}
, , et
... .
L
A latti
e is join-in(cid:28)nite distributive (JID) if
a∧ B = (a∧B)
(2.7)
(cid:16)_ (cid:17) _
whenever the supremum on the left hand side exists. The dual
ondition is
alled
meet-in(cid:28)nite distributive (MID). Note that distributivity does not imply either of
these properties, even for
omplete latti
es (see e.g. [DP02℄). Also note that every
Boolean algebra is both join-in(cid:28)nite and meet-in(cid:28)nite distributive (see e.g. [Kop89,
a
Ch. 1, Ÿ1℄). We will need to re
all that
an be repla
ed with a supremum:
L A ∧ B = (A∧B) A
Proposition 2.23. If is (JID), then whenever
B
and both exist. Similarly for the (M(cid:0)IWD).(cid:1) (cid:0)W (cid:1) W W
W (M,·) · ∨
De(cid:28)nition 2.24. We shall
all a monoid in(cid:28)nitely left -distributive over
(·,∨)
or in(cid:28)nitely left -distributive if
a· B = (a·B) a∈ M B ⊆ M
(2.8) for all and ,
_ _
B (·,∨)
whenever exists, and
all it in(cid:28)nitely right -distributive if
W A ·b = (A·b) A⊆ M b ∈ M
(2.9) for all and ,
A (cid:16)_ (cid:17) _ (·,∨)
whenever exists, and we
all the monoid in(cid:28)nitely -distributive if it is
(·,∨) (·,∧)
both in(cid:28)niWtely left and right -distributive. The de(cid:28)nition of in(cid:28)nite -
distributivity is exa
tly analogous.
(M,·) (·,∨) A · B =
Proposition 2.25. If is in(cid:28)nitely -distributive, then
(A·B) A B (·,∧)
whenever and both exist; similarly for in(cid:28)nite (cid:0)W-di(cid:1)str(cid:0)ibWutiv(cid:1)ity.
W W W