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3.13.
MONOID OF DISTRIBUTIONS OF BINARY FORMULAS
221
that the formulas
θ
v1,u
(a, y),θ
3
u1,u2,u
(a, y)
3
, and
θ
u1,v
(a, y)
2
wise equivalent, i. e.,
dv1, u3e = du1, u2, u3e = du1, v2e. ¤
In view of associativity, using the induction on the number of
brackets, we prove that all operations
P(ρ
P(ρ
ρ
ν(p)
where
ν(p)
ν(p)
)\{∅}
) \ {∅}
are generated by the binary operation
and the values
dX1, X2, . . . , Xke,X1, X2, . . . , Xk⊆
, do not depend on the sequence of adding of brackets for
X
i,i+1,...,i+m+n
X
1,2,...,k
= dX1, X2, . . . , Xke
Thus the structure
dX
SI
i,i+1,...,i+m
hP(ρ
ν(p)
group admitting the representation of all operations
terms of the language
d·, ·e
. Below the operation
noted also by·and we shall use the record
Since by the choice of the label0for the formula
equalities
the semigroup
X · {0} = X
SI
ν(p)
and
{0} · X = X
has the unit
d·, ·, . . . , ·e
, X
i+m+1,i+m+2,...,i+m+n
.
) \ {∅}; d· , ·ei
ν(p)
uv
are true for any
{0}
, and it is a monoid. We
acting on sets in
d·, ·e
on the set
is a semi-
d·, ·, . . . , ·e
d·, ·e
will be de-
instead of
u · v
(x ≈ y )
X ⊆ ρ
have
Y · Z =[{yz | y ∈ Y, z ∈ Z}
are pair-
e,
by
.
the
ν(p)
,
for any sets
Y, Z ∈ P(ρ
ν(p)
) \ {∅}
in this structure.
Thus the following proposition holds.
3.13.0.2. Proposition.
S1(T )
SIp(·, ·, . . . , ·)
of the monoid
, and the regular labelling function
on the set
SI
ν(p)
3.13.0.3. Denition.
For any complete theoryT, any type
ν(p)
, any operation
P(ρ
ν(p)
) \ {∅}
interpretable by a term
.
The monoid
SI
is called the
ν(p)
monoid of
binary semi-isolating formulas over the labelling function
the
SI
-monoid
ν(p)
.
In view of Propositions 3.10.0.4 and 3.13.0.1 we obtain
ν(p)
p ∈
or
222
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.13.0.4. Proposition.
p ∈ S1(T )
≤0
SI
ν(p)
set
U≤0(
, and the regular labelling function
(
respectively
U≥0,
SI
U≥0∪ U0)
By Proposition 3.12.0.15, the
SI
coincides with the
ν(p)
restrictions of monoids
≤
groupoid
3.14.
0
P
and the monoid
ν(p)
α
-deterministic and almost
α
-deterministic
For any complete theoryT, any type
ν(p)
, the restriction
SI
ν(p)
.
ν(p)
. Besides, the
,
SI
≥0,neu
ν(p)
)
≥0
ν(p)
is a submonoid of
(1, 2)
I
-groupoid
ν(p)
≤0
SI
ν(p)
and
P
≥
ν(p)
SI
SI
0
ν(p)
.
of the monoid
SI
-restriction of the monoid
P
ν(p)
≥0
equal respectively to the
ν(p)
-monoids
In the following denition, we generalize the notions of deter-
ministic and almost deterministic structure
P
dened in Section
ν(p)
3.4.
3.14.0.1. Denition.
α
be a positive ordinal, and
ρ
ν(p),α,n
{u ∈ ρ
Let
U0be a subalphabet of the alphabetU,
n ≥ 1
| si(u) ≤ α, deg(u) < n
ν(p)
be a natural number. We put
for
si(u) = α},
[
ρ
ν(p),α
n∈ω
ρ
ν(p),α,n
.
to the
(1, 2)
-
The partial subalgebra
called
the set
(α, n)
-deterministic
du1, u2e ∩ U0consists of labels having thesi-ranks
SI
ν(p)
if for any labels
contains less thannpairwise non-
The partial subalgebra
calledα-deterministic
The partial subalgebra
called
bels
having thesi-ranks
almostα-deterministicor(α, ω)
u1, u2∈ ρ
ν(p),α
≤ α
∩ U0, the set
SI
ν(p)
if
SI
ν(p)
SI
¹ U0is
ν(p)
and contains nitely many pairwise non-
∼α-equivalent labels ofsi-rankα.
¹ U0of the monoid
u1, u2∈ ρ
SI
ν(p),α,n
≤ α
ν(p)
∩ U0,
and
is
∼α-equivalent labels ofsi-rankα.
¹ U0of the monoid
(α, 2)
-deterministic.
¹ U0of the monoid
-deterministic
SI
SI
ν(p)
ν(p)
is
is
if for any la-
du1, u2e ∩ U0consists of labels
3.14.α-DETERMINISTIC
By
the denition, each
U0is a union of its
eachα-deterministic structure
If
U0= U
we shall not point out restrictions to the set
SI
-MONOIDS
ν(p)
(α, ω)
-deterministic structure
(α, n)
-deterministic substructures,
SI
¹ U0is almostα-deterministic.
ν(p)
SI
n ≥ 1
considering structures.
Below we show some basic properties of (almost)α-deterministic
partial algebras
SI
ν(p)
¹ U0.
223
ν(p)
. So
U0for
¹
3.14.0.2. Proposition
is(almost)α
structure
Proof
-deterministic andβis a positive ordinal then the
(SI
ν(p)
¹ U0) ¹ β
is obvious.
3.14.0.3. Proposition.
α, β
, where
α, β > 0,β ∈ ω + 1
(Monotony).
is also(almost)α
¤
For any monoid
If a structure
, the following conditions are
equivalent:
(1)
the monoid
(2)
si(u1◦ u2) ≤ α
u2) = α
then
Proof.
The implication
(2) ⇒ (1)
by the hypothesis,
v ∈ du1, u2e
, so
si(v) = si(u1◦ u2) = α
monoid
SI
ν(p)
SI
deg(u1◦ u2) < β
is
ν(p)
(α, β)
for any labels
.
(1) ⇒ (2)
-deterministic;
u1, u2∈ ρ
is obvious.
. Consider arbitrary labels
si(u1◦ u2) ≤ α
du1, u2e
then
is
(α, β)
-deterministic.
consists of labels ofsi-ranks
deg(v) ≤ deg(u1◦ u2) < β
and
¤
u1, u2∈ ρ
v E (u1◦ u2)
Proposition 3.14.0.3 immediately implies
3.14.0.4. Corollary.
For any monoid
SI
ν(p)
nalαthe following conditions are equivalent:
(1)
the monoid
(2)
si(u1◦ u2) ≤ α
SI
is almostα-deterministic;
ν(p)
for any labels
u1, u2∈ ρ
SI
ν(p)
-deterministic.
SI
ν(p),α,β
and ordinals
ν(p)
and if
ν(p),α,β
si(u1◦
. Since,
for any label
≤ α
, and if
. Thus, the
and a positive ordi-
.
ν(p),α
¹ U
0
3.14.0.5. Corollary.Ifsi(p)
is almost
3.14.0.6. Proposition.
tic then the structure
si(p)
-deterministic.
If a monoid
SI
ν(p),α
terministic monoid.
is an ordinal then the monoid
SI
SI
ν(p)
ν(p)
¹ α
is
(α, β)
-determinis-
is also an
(α, β)
SI
ν(p)
-de-
224
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Proof.
ence of unit
enough to note that for any labels
a labelvin
since, by the hypothesis,
Since for anyα-restriction the associativity, the pres-
{0}
SI
, and the
ν(p),α,β
belonging
(α, β)
-determinacy is preserved, it is
u1and
du1, u2e
si(u1◦ u2) ≤ α
u2in
SI
ν(p),α,β
. We can take
and if
si(u1◦ u2) = α
u1◦ u2for
deg(u1◦ u2) < β.¤
3.14.0.7. Proposition.
SI
ν(p)
is
si(p)
-deterministic if and only if the value
If
si(p)
is an ordinal then the monoid
deg(p)
dened or equals1.
Proof.
and can not be achieved by labels in
u1, u2∈ ρ
If
deg(p) ≥ 2
ofsi-rankα. Then
whence the monoid
then there is unique, up to
If
, the set
ν(p)
deg(p)
is not dened the ordinal
du1, u2e
then there are non-
du1∨ u2, 0e
SI
ρ
. In particular, for any
ν(p)
does not contain labels ofsi-rankα.
∼α-equivalent labels
contains the labels
is notα-deterministic. If
ν(p)
∼α-equivalence, label in
α = si(p)
u1, u2∈ ρ
deg(p) = 1
ρ
thesi-rankα. Since such a label is unique, the monoid
α
-deterministic.
3.14.0.8. Proposition.
ministic if and only if the structure
¤
The structure
SI
P
ν(p)
ν(p),1,2
is(almost)deter-
is(almost)1
ministic.
there is
is not
is limit
u1and
having
ν(p)
SI
-deter-
then
ν(p)
v
ν(p)
u2,
is
Proof
3.14.0.9. Proposition.
T,ν(p)
follows by the equality
Let
SI
ν(p),1,2
p(x)
be a complete1-type of a theory
be a regular labelling function, and
= P
si(p) < ω
conditions are equivalent:
(1)
the monoid
(2)
the set
(3)
the set
(4)
the set
Proof.Ifsi(p) > 1
and so the set
disjunction of labels in
the label
u1∨. . .∨unbelongs also to
and the monoid
SI
ρ
is nite;
ν(p)
ρ
ν(p),1
ρ
ν(p),1,2
ρ
is also innite. Since each label in
ν(p)
SI
ν(p)
is
(1, n)
ν(p)
-deterministic for some
nite;
(
consisting of all atoms
then, by
ρ
ν(p),1,2
is not
si(p) < ω
and for any labels
ρ
ν(p),1
(1, n)
-deterministic for
, the set
, the set
none of the conditions (1){(4) is not satised.
.
¤
ν(p)
. The following
u ∈ ρ
ν(p)
ρ
ν(p),1
)
is nite.
is innite
ρ
u1, . . . , un∈ ρ
ρ
ν(p),1,2
is innite
n ∈ ω
n ∈ ω
ν(p),1
. Thus,
;
is a
ν(p),1
3.14.α-DETERMINISTIC
If
si(p)
= 1
then each label in
represented as a disjunction of labels in
ditions (2){(4) are equivalent. If the set
labels then there are
monoid
SI
ν(p)
is
(1, 2m−1)
nite then, for pairwise distinct labels
du1∨ . . . um, 0e
the monoid
contains
SI
ν(p)
2m−1
is not
SI
-MONOIDS
ν(p)
ρ
has thesi-rank1and is
ν(p)
ρ
ρ
labels forming the set
-deterministic. If the set
u1, . . . , um∈ ρ
2m− 1
labels and, sincemis not bounded,
(1, n)
-deterministic for anyn. Thus, the
. Thus, the con-
ν(p),1,2
ν(p),1,2
contains
ρ
condition (1) is equivalent to each of the conditions
Proposition 3.14.0.9 and Corollary 3.7.0.6 imply
. Hence, the
ν(p)
ρ
ν(p),1,2
, the set
ν(p),1,2
(2){(4).¤
225
m ∈ ω
is in-
3.14.0.10. Corollary.
T,ν(p)
a
(1, n)
be a regular labelling function,
-deterministic monoid, for some
label. Then the groupoid
Let
SI
p(x)
be a complete1-type of a theory
ν(p),1,2
si(p) < ω
n ∈ ω
generates the strict order prop-
, and
, having a negative
SI
erty.
Lemma 1.3.1.3 and Proposition 3.14.0.9 imply
3.14.0.11. Corollary.
T,ν(p)
is a regular labelling function, and all
principal, then the monoid
n ∈ ω
.
If
p(x)
is a complete1-type of a theory
(2, p)
-types arep-
SI
ν(p)
is
(1, n)
-deterministic for some
By Corollaries 3.14.0.10 and 3.14.0.11, we obtain
3.14.0.12. Corollary.
T,ν(p)
(2, p)
be a regular labelling function,
-types bep-principal. Then the groupoid
Let
p(x)
be a complete1-type of a theory
ρ
∩ U−6= ∅
ν(p)
SI
ν(p),1,2
, and all
generates
the strict order property.
For a type
a model
p(x)
MofT
and a positive ordinalα, we denote by
SI
) the relation of semi-isolation (over∅) on a set
of realizations ofprestricted to the set of formulas ofsi-rank
SI
{(a, b) | M |= p(a) ∧ p(b)
p,α
andasemi-isolates
ν(p)
p,α
b
(in
≤ α
is
:
and
Clearly,
SI
Ip= SI
ν(p),1,2
sition 3.4.0.5.
by a formula
for any type
p,1
= P
the following proposition generalizes Propo-
ν(p)
θu(x, y)
with asi-rank
p ∈ S1(∅)
. Seeing this equality
≤ α}.
226
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.14.0.13. Proposition.
T,ν(p)
be a regular labelling function, andαbe a positive ordinal.
Let
p(x)
be a complete1-type of a theory
The following conditions are equivalent:
(1)
the relation
SI
(
on a set of realizations ofpin any model
p,α
M |= T)is transitive;
(2)
the structure
SI
θ
p,α
u
2
Proof.
and
(b, y)
Leta,b, andcbe realizations ofpsuch that
(b, c) ∈ SI
respectively. If the structure
deterministic monoid then
to
SI
by the semi-isolating formula
p,α
a,b
, andcare arbitrary we have
Assume now that for some
contains a labelusuch that
SI
p,α
is an almostα-deterministic monoid.
ν(p),α
by semi-isolating formulas
si(u1◦u2) ≤ α
(2) ⇒ (1)
u1, u2∈ ρ
si(u) > α
set
q(a, y) {θ
is consistent, where
that
|= θ
(a, b) ∧ θ
u
1
and
(b, c) ∈ SI
the relation
holds.
¤
(a, y)} ∪ {¬θv(a, y) | v ∈ SIp(u1, u2), si(v) ≤ α}
u1,u
2
|= p(a)
u
but
p,α
SI
is not transitive and the implication
p,α
. Consider realizationsbandcofpsuch
(b, c)
and
2
(a, c) /∈ SI
|= q(a, c)
by the construction ofq. Thus,
p,α
(a, b) ∈
θ
SI
ν(p),α
and the pair
θ
(x, y)
u1,u
2
(a, y)
u
1
is an almostα-
(a, c)
. Since elements
and
belongs
.
ν(p),α
the set
SIp(u1, u2)
. Then by compactness the
. We have
(a, b) ∈ SI
p,α
(1) ⇒ (2)
Note that for any ordinal
α > 0
there are no
(p, θu, p)
linking distinct realizations ofpand satisfying the conditions
si(u) ≤ α
antisymmetric. Since
, and
si(u−1) ≤ α
SI
p,α
, if and only if the relation
is reexive, the denition of
ν(p)
Propositions 3.10.0.4, 3.14.0.13 imply
3.14.0.14. Corollary.
T,ν(p)
be a regular labelling function, andαbe a positive ordinal.
Let
p(x)
be a complete1-type of a theory
The following conditions are equivalent:
(1)
the relation
ofpin any model
(2)
the structure
and
ρ
ν(p),α
⊆ U≤0.
This partial order
If
SI
is not identical, it has innite chains.
p,α
SI
p,α
M |= T
SI
ν(p),α
SI
ν(p),α
is a partial order on a set of realizations
;
is an almostα-deterministic monoid
is identical if and only if
ρ
ν(p),α
-edges,
u > 0
SI
p,α
and
= {0}
,
is
.
3.14.α-DETERMINISTIC
Prop
ositions 3.10.0.4 and 3.14.0.13 also imply
SI
ν(p)
-MONOIDS
227
3.14.0.15. Corollary.
T,ν(p)
be a regular labelling function, andαbe a positive ordinal.
Let
p(x)
be a complete1-type of a theory
The following conditions are equivalent:
(1)
the relation
realizations ofpin any model
(2)
the structure
and consists of labels in
Since each semi-isolating formula
lutions is equivalent to a disjunction of isolating formulas
SI
is an equivalence relation on the set of
p,α
SI
M |= T
is an almostα-deterministic monoid
ν(p),α
;
U≥0.
θu(a, y)
with nitely many so-
θ
u
(a, y)
i
each almost deterministic element has thesi-rank1and so belongs
to the set of labels in the structure
number of solutions for
θu(a, y)
. In particular, each deterministic
element belongs to the set of labels in the structure
SI
ν(p),1,n+1
, wherenis the
SI
ν(p),1,2
.
It is shown in Proposition 3.4.0.9 that if elementsuandvare
(almost) deterministic then each element
v0in
u · v
is also (almost)
deterministic. Hence, thesi-rank1is preserved for compositions
u ◦ v
si
of (almost) deterministic elementsuandv. Moreover, the
-degree1is preserved for compositions of deterministic elements.
In Figure 3.2, the fragments of Hasse diagram are presented
illustrating the links of the structure
above, being restrictions of
superscripts
sets
U≤0and
·≤0and
·≥0point out on restrictions of
U≥0respectively, and the subscripts to the upper
SI
SI SI
with structures
ν(p)
to subalphabets ofU. Here the
SI
to the
estimates forsi-ranks andsi-degrees of labels. In Figure 3.2, a, a
hierarchy of structures
SI
,
α ≤ si(p)
α
, is depicted starting with the
trivial substructure; in Figure 3.2, b, links between substructures of
SI
of
SI
the Hasse diagram for substructures of
presented diagrams for
are presented; in Figure 3.2, c, links between substructures
ν(p),1
for
α+1
1 ≤ α < si(p)
are shown. For a limit ordinal
SI
is obtained by union of
β
α < β
. If an ordinal
β ≤ si(p)
β ≤ si(p)
is not limit,
the Hasse diagram corresponds to the union of presented diagrams
for
α < β
with the removal of structures
SI
≤0
β+1,2
and
SI
≥0
β+1,2
.
,
,
228
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
≤0
2,2
¡
≤0
1
≤0
1,3
≤0
1,2
@
¡
¡
¡
@
¡
¡
¡
@
•
•
•
•
b
·
·
·
·
·
·
·
@
@
@
¡
SI
SI
SI
¡
{0}
@
@
@
¡
1
@
1,3
@
1,2
@
@
@
@
¡
SI
•
SI
•
·
·
·
·
·
·
·
•
SI
3
¡
¡
•
SI
·
·
·
·
·
·
¡
·
¡
¡
•
SI
2
•
SI
¡
¡
¡
•
SI
1
•
SI
@
@
•
{0}
a
•
@
•
·
·
·
·
·
·
·
@
•
@
•
¡
Figur
SI
SI
SI
SI
≥0
2,2
≥0
1
≥0
1,3
≥0
1,2
e 3.2
≤0
α+2,2
¡
¡
≤0
α+1
¡
≤0
α+1,4
¡
≤0
α+1,3
¡
≤0
α+1,2
¡
¡
¡
¡
SI
α+1
•
@
·
·
@
·
@
·
·
@
·
·
SI
•
α+1,4
@
@
@
@
SI
•
α+1,3
@
@
@
@
SI
•
α+1,2
@
@
@
SI
@
α
•
SI
•
¡
¡
•
SI
·
·
·
·
·
·
¡
·
¡
¡
•
SI
¡
¡
¡
•
SI
¡
¡
¡
•
SI
c
@
@
@
@
≥0
SI
α+2,2
•
≥0
•
SI
α+1
·
·
·
·
·
·
·
≥0
•
SI
α+1,4
≥0
•
SI
α+1,3
•
≥0
SI
α+1,2
3.15.
In this Section, we shall consider both the monoids
POSTC
-monoids
SI
ν(p)
and
their expansions (with the addition of empty set∅to the universe
such that
and relations of
X · ∅ = ∅ · X = ∅
POSTC
-algebras containing these monoids. These
for
X ∈ P (ρ
)
) by operations
ν(p)
expansions
M
are called
preordered monoids with relative set-theoretic operations
and compositions
POSTC
-monoids
ν(p)
We collect basic structural properties of
and show that any expanded monoid
list of properties, coincides with some
ν(p)
hP(ρ
); ·, E, ∨, ∧, (· ∧ ¬ ·), ◦i
ν(p)
over regular labelling functions
.
SI
, satisfying the following
POSTC
ν(p)
POSTC
-monoid
ν(p)
, or briey
-monoids
ν(p)
M
ν(p)
.
3.15.
POSTC
-MONOIDS
229
−
3.15.0.1.
Denition.
bet consisting of a set
elements
write
and
u < 0
u · v
, a set
U0of
for any element
instead of
{u} · {v}
Let
U = U
U−of
negative elements
neutral elements
u ∈ U−,
considering an operation·on the set
˙
∪ {0}˙∪ U
, and zero0. As above, we
u > 0
P(U);U≤0 U−∪ {0},U≥0 U+∪ {0}
A structure
POSTC
{0}
elements
-monoid
•
the operation·of the monoid
is generated by the function·on elements inUsuch that each
u, v ∈ U
X, Y ∈ P(U ) \ {∅}
M = hP(U); ·, E, ∨, ∧, (· ∧ ¬ ·), ◦i
if it satises the following conditions:
hP(U) \ {∅}; ·i
dene a nonempty set
the following equality holds:
+
˙
∪ U0be an alpha-
, a set
U+of
for any element
.
is called a
with the unit
(u · v) ⊆ U
: for any sets
positive
u ∈ U+,
X · Y =[{u · v | u ∈ X, v ∈ Y };
if
X ∈ P(U )
the relationEon the set
element∅; this preorder is induced by the partial order
then
X · ∅ = ∅ · X = ∅
P(U)
;
is a preorder with the least
0
E
on the
setUof labels (forming a upper semilattice) by the following rule:
if
X, Y ∈ P(U )
u ∈ X
there is a label
there is a label
then
u ∈ X
X E Y
v ∈ Y
with
if and only if
with
u E0v
u E0v
;
X = ∅
, or for any label
and for any label
v ∈ Y
•
a label
any label
label0is an atom; some labels in
moreover, if only labels
there are no greatest labels among labelsv; only labels in
over each label in
•
the operations
u ∈ U
v ∈ U
is called an
; only labels in
v ∈ U≥0lay under a label
U0;
∨, ∧, (· ∧ ¬ ·)
atomifv E u
−
˙
U
∪ {0}˙∪ U+may be atoms; the
implies
v = u
U≥0lay under each label in
u ∈ U0then
on the set
U ∪ {∅}
form a dis-
for
U0,
U0lay
tributive lattice with relative complements, moreover, for any ele-
ments
u, v ∈ U ∪ {∅}
,
u E0v ⇔ u ∧ v = u ⇔ u ∨ v = v,
(u ∧ ¬v) = ∅ ⇔ u E v;
•
the operation◦is dened on the setUsuch that for any labels
u, v ∈ U
the label
u ◦ v
is the greatest element of the set
u · v
;
230
corresponding operations on the set
τ ∈ {∨, ∧, ◦}
(· ∧ ¬ ·)
operation on the set
{u ∧ ¬v | u ∈ X, v ∈ Y }
operations
∨
; if
rank of semi-isolation
deg(u)
as well as equivalence relations
X ∈ {U, U ∪ {∅}}
of
to sets of labels ofsi-degree
Chapter
•
the operations
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
∨, ∧, ◦
on the set
P(U)
U ∪ {∅}
then
X τ Y = {u τ v | u ∈ X, v ∈ Y }
on the set
P(U)
is also induced by the corresponding
U ∪ {∅}
: if
X, Y ∈ P(U)
;
•
the sets
u ∈ U0and
•
repeating the denition in Section 2, for each label
U−∪{∅}
∨, ∧, (· ∧ ¬ ·)
v ∈ U
and
U≥0∪{∅}
; the set
then
(u ∨ v) ∈ U0;
si(u) ≥ 1
are closed with respect to the
U0is closed under the operation
and the
of labeluis dened inductively,
degree of semi-isolation
si(∅) = 0,deg(∅) = 1
∼α, restrictions
and restrictions
M
to sets of labels ofsi-ranks
•
the restriction
hP(U)\{∅}; ·i
< β
;
0
M
,
M
α
≤ α
, and for labels ofsi-rank
of the monoid
1,2
is aI-groupoid;
are induced by the
: if
X, Y ∈ P(U )
; the operation
then
X ∧ ¬Y =
u ∈ U
Xα,
X
0
for restrictions
α,β
α,β
hP(U)\{∅}; ·i
and
, the
of sets
M
,
0
α
•ifu < 0
for any
v ∈ U
•ifu > 0
•ifu, v ∈ U≥0∪ U0, and
•
for any element
elements
and
v ∈ U+then
•
if a positive elementubelongs to a set
U+, then
By the denition each
submonoids
P(U+∪ {0})
U0= ∅
3.15.0.2. Denition.
for any label
then sets
u · v
;
and
v > 0
u > 0
u0> 0
such that
u−1⊆ v−1;
−1
u−1⊆ v
≤0
M
2
and
· v
−1
1
M
respectively, being also
and
U−∪ U0= ∅
u ∈ U
there is an atom
and
then
(u · v) ⊆ U≥0;
u ∈ U0or
there is a nonempty set
0 ∈ (u· u0) ∩ (u0· u)
.
POSTC
≥0
with the universes
respectively).
A
POSTC
v · u
consist of negative elements
v ∈ U0, then
(u · v) ⊆ U0;
u−1of
; in this case if
v1· v2, where
-monoid
M
contains
P(U−∪ {0})
POSTC
-monoid
v ∈ U
-monoids (with
M
is called
such that
v E u
inverse
u E0v
v1◦ v2∈
POSTC
and
U+∪
atomic
.
-
if
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