Positive semigroups are a particularly important class of semigroups, since the underlying relation is a partial order, and so the structure more closely aligns with the standard reliability setting. As usual, our starting point is a measurable space . Semigroups are assumed to be measurable and satisfy the left cancellation property, as discussed in Section 1.
Definitions and Properties
Suppose that is a semigroup.
- is a strict positive semigroup if for every .
- is a positive semigroup if has an identity element and is a strict positive semigroup where .
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Recall that a semigroup is assumed to satisfy the left cancellation law. Note that a strict positive semigroup cannot have an idempotent element and hence cannot have an identity (left, right or two-sided) or a zero (left, right, or two-sided).
A strict positive semigroup can be made into a positive semigroup with the addition of an identity element.
Suppose that is a strict positive semigroup. Let be an element not in and define . Extend to by the rule that for all . Then is a positive semigroup.
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This follows from the more general result in Section 1 since does not have a left identity. In terms of the measure theory, suppose that is the underlying measure space. We define by adding to all sets of the form where .
Note that the algebraic assumptions of a positive semigroup do not rule out the possibility that for some with . A positive semigroup has no nontrivial inverses.
Suppose that is a positive semigroup with identity . Then if and only if
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Trivially if then since is an identity. If and then since is algebraically closed. If and then . Finally if and then .
If is a positive semigroup then a product over an empty index set is interpreted as the identity . In particular, for .
Suppose that is a positive semigroup. The positive sub-semigroup generated by has base set . This semigroup is isomorphic to the standard discrete semigroup , with isomorphism .
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Suppose that with and that . Then and so by left cancelation, . But then by repeated applications of [3], a contradiction. Hence the mapping is one-to-one and onto. Of course also for .
Graphs
The following result is the reason that positive semigroups have special importance.
Suppose that is a positive semigroup with identity , and that is a sub-semigroup.
- If then the relation associated with and is a partial order.
- If then the relation associated with and is a strict partial order.
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Since is algebraically closed, the relation associated with and is transitive from Section 1.
- Suppose that . Then since for . Hence is reflexive. If and for some then there exists with and . Hence . By the left cancellation property, and so from [3], . Thus and so is anti-symmetric.
- Suppose that . If for some then there exists with . By left cancellation again, , a contradiction. Hence is anti-reflexive, and since it's transitive, also asymmetric.
The most important case is when and as usual, we drop the subscript so that the relation associated with is a partial order. In the case that , the associated relation is the corresponding strict partial order , so again we drop the subscript. At the other extreme, when , the associated relation is equality . Of course all of the definitions and results in Section 2.1 on partial order graphs apply to but we will repeat some of these in terms of the semigroup operator. Note first that the identity is the minimum element of since for every .
Suppose again that is a positive semigroup and that . Then relative to the graph ,
- is increasing if and only if and imply , or equivalently implies .
- is decreasing if and only if and imply .
- is convex if and only if , and imply .
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All three results follow form the definitions, since if and only if for .
Suppose that is a positive semigroup with associated partial order and that is another partial order graph. Suppose also that . Then relative to the partial orders,
- is increasing if for .
- is decreasing if for .
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Again, the results follow from the definitions since for .
Suppose that is a positive semigroup with associated partial order . By the result in Section 1, for fixed , the mapping is an isomorphism from the graph to the graph .
Irreducible Elements
In this subsection, we assume that is a discrete positive semigroup with associated partial order .
An element is irreducible if cannot be factored for , except for the trivial factoring .
The covering graph of can be characterized in terms of the irreducible elements.
If then covers if and only if for some irreducible element . That is, the covering relation is the relation associated with , the set of irreducible elements of .
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Suppose that covers . Then so there exists with . If for some then which would be a contradiction. Thus, has no non-trivial factorings and hence is irreducible. Conversely, suppose that for some irreducible element . Then . Suppose there exists with . Then and for some . Thus so by left cancellation, . But this is a contradiction since is irreducible, and so covers .
Suppose that is left finite with as the set of irreducible elements. Then can be factored finitely over for every . That is, for some where for .
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Recall that left finite means that the corresponding partial order graph is well founded, and hence does not have an infinite descending chain. Hence for , there exists a finite path in the covering graph from to , say where , , , and covers for each . But then for each where . Hence .
Of course, the factoring of over is not necessarily unique, and different factorings of over may have different lengths. If a partial order graph is associated with a positive semigroup, then is uniform if and only if, for each , all factorings of over (the set of irreducible elements) have the same length.
Suppose again that is left finite. Then .
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Suppose that is a homomorphism from into and that for each . If , then from [10], can be factored over so that where for each . But then
Hence is a critical set and so
We can certainly have . The semigroup corresponding to the subset partial order on finite subsets of studied in Chapter 9 has infinitely many irreducible elements but the semigroup dimension is 1. For a positive semigroup, there are two definitions for dimension, one corresponding to the semigroup itself, and one corresponding to the corresponding graph, leading to an interesting problem:
Möbius Inverstion
In this subsection we assume that is a discrete, left-finite positive semigroup with identity . So the theory of Möbius inversion in Section 1.2 applies to the associated partial order graph .
Let denote the Möbius kernel of . Then for .
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The fact that is left finite means that the Möbius kernel is well defined and that is well founded so that we can use partial order induction. Let . First, if and only if so it follows that if then . Next, . Suppose now that . Recall that maps one to one and onto . For the induction hypothesis, suppose that for all . Then
The result is to be expected because of the self-similarity property of a positive semigroup. It follows that the general Möbius kernel can be obtained from a simpler function.
The Möbius function of is defined by for , and is defined inductively as follows:
- for
The Möbius kernel is related to the Möbius function by
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Suppose that so that makes sense. From [13],
Groups
Often (but not always) the positive semigroups that are of interest to us are embedded in groups. Suppose that is a group and that is a positive, sub-semigroup of . We can think of as a set of positive elements of and as the corresponding set of strictly positive elements of . Often we are interested in a maximal set of positive elements in a sense.
Suppose that is a group. A positve, sub-semigroup is maximal if is not a sub-semigroup of a proper subgroup of .
In the case of a commutative group, a positive sub-semigroup is maximal if and only if we can write each element of the group as the difference between positive elements.
Suppose that is a commutative group and that is a positive sub-semigroup. Let . Then
- is a subgroup of containing .
- is maximal if and only if .
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- If so that and where then and . Hence is a semigroup of . Moreover, contains since for where is the identity.
- If is maximal then by definition. Conversely, every subgroup of containing must also contain . Hence if then is maixmal.
Examples
Many of the semigroup examples in Section 1 are positive semigroups.
The space is a maximal positive sub-semigroup of the commutative group . The associated order is the total order .
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Trivially we can write as where . In fact, either or .
The space is a maximal positive sub-semigroup of the commutative group . The associated order is the total order .
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Trivially we can write as where . In fact, either or .
The group and semigroup in [18] are isomorphic to the group and semigroup in [19] with isomorphism from onto . These semigroups and other related spaces are studied in Chapter 3.
The space is a maximal positive sub-semigroup of the commutative group . The associated order is the total order .
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Trivially we can write as where . In fact, either or .
The semigroup and other related spaces are studied in Chapter 4.
The space is a positive semigroup where is ordinary multiplication. The associated partial order is the division relation.
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That is, if and only if divides so that for some .
The semigroup and other related spaces are studied in Chapter 6.
The free semigroup over a countable alphabet is a positive semigroup. The associated partial order is defined by if and only if is a prefix of .
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Recall that consists of finite words over (strings of letters from ). The semigroup operation is concatenation. The empty word with no letters is the identity. The free semigroup over can be embedded in the free group over by adding the inverse letters
for and extending the operation in the obvious way so that for . Of course, [17] does not apply since is not commutative. Nonetheless, is clearly maximal since any subgroup of that contains would have to contain the set of inverse letters, which would then imply .
The free semigroup and other related spaces are studied in Chapter 5.