In this section we consider sub-semigroups of the standard discrete space that was studied in Section 1. Of particular interest are numerical semigroups defined in [4]
Basics
Throughout this section we assume that is a sub-semigroup of with the properties that
- for some .
Details:
Of course, the meaning of sub-semigroup is closure under the semigroup operation, so if then .
The first condition means that is a positive sub-semigroup of , and the second condition rules out the trivial case . Of course, the partial order associated with is given by if and only if for some . So implies for but not conversely, in general. That is, is not necessarily algebriacally complete. The partial order graph is left finite and hence the associated -algebra is , the collection of all subsets of . A standard way to construct a subsemigroup is by means of a generating set.
If , then the sub-semigroup generated by is the interesection of all subsemigroups of that contain (and the identity element 0). Equivalently, the base set of the sub-semigroup generated by is
where as usual, an empty sum is interpreted as . The generating set is minimal if no proper subset of generates the same semigroup.
If is a sub-semigroup of then the set of irreducible elements of is the unique minimal generating set of .
Details:
The semigroup is left finite, so by a result in Section 2.2, if then for some and . That is, . Clearly is also miniimal. Suppose that is a proper subset of and that . But then since is irreducible in . Conversely, if is a generating set for then clearly .
A numerical semigroup is a sub-semigroup of that contains all but finitely many elements of .
is a numerical semigroup if is finite.
- is the set of gaps.
- is the genus of the semigroup.
- is the Frobenius number of the semigroup.
Details:
In part (c), the maximum is with respect to the ordinary order , of course.
So if is the Frobenius number of then . Sets that generate numerical semigroups have some special properties.
A semigroup generated by a set is a numerical semigroup if and only if .
Suppose that . The is a multiple of for every and hence is a multiple of for every . So the set of gaps is infinite.
Let denote the set of irreducible elements of the numerical semiegoupr (so that is the minimal generaing set). Then
- is finite
- is the embedding dimension of the semigroup.
- is the multiplicity of the semigroup.
Details:
In part (c), the minimum is with respect to the ordinary order .
Suppose again that is a sub-semigroup of and that is the set of irreducible elements. As usual, we let denote the covering graph corresponding to the partial order graph . Then for , if and only if for some .
Probabillity
Once again, suppose that is a sub-semigroup of , as defined in [1]. Since the corresponding partial order graph is a lattice, the graph is stochastic. That is, the reliability function of a probability measure on uniquely determines . As usual, our primary interest is in exponential distributions for and constant rate distributions for the corresponding partial order graph and for its covering graph .
is the reliability function of an exponential distribution on if and only if there exists such that for . The rate constant is .
Details:
Suppose that and that for . Then trivially is multiplicative: for . Also,
so is the reliability function of an exponential distribution on by a result in Section 2.3. The rate constant is given by . Conversely, suppose that is the reliability function for an exponential distribution on .Let . If then so by the memoryless property, so . That is, has the form given in the theorem with .
The following result is essentially a restatment of [7].
Suppose that has a geometric distribution on . Then the conditional distribution of given has an exponential distribution on . Moreover, every exponential distribution on has this form.
Details:
Suppose that has the geometric distribution on with success parameter . Then of course, has an exponential distribution on the standard discrete semigroup , with reliability function defined by for . By a standard result in Section 2.4, the conditional distribution of given is exponential for the sub-semigroup and the reliability function of this conditional distribution is simply restricted to . Conversely, an exponential distribution on has a reliability function of the form for where . The corresponding density function is for where the rate constant is simply the normalizing constant. Equivalently, if has the geometric distribution on with success parameter then for .
Of course, the exponential distribution described in [7] has constant rate for the associated partial order graph . The distribution also has constant rate for the covering graph .
Suppose that random variable in has density function given by for , with parameter , and where is the normalizing constant. Then has constant rate for the covering graph . Moreover, every constant rate distribution for has this form.
Details:
As before, let denote the set of irreducible elements of , or equivalently, the minimial generating set. Suppose that has density given in the theorem. Then the reliability function for is given by
Conversely, suppose that is the density function of a distribution on with constant rate for . Let denote the reliability function. Then
The characterstic equation for this linear recurrence relation is
or equivalently where for .
Note that , , and for . So if , then there exists a unique distribution with constant rate , with reliability function of the form for . Of course, depends on .
Examples
Fix and let . The numerical semigroup has (minimal) generating set . The genus and Frobenius number are , and the embedding dimension and multiplier are . The exponential distribution on with parameter as defined in [7] has probability density function given by
so the rate constant is . This distribution also has constant rate for the covering graph , with rate constant. .
Fix an odd integer and let . The numerical semigroup has embedding number 2, multiplier 2, Frobenius number and genus . The exponential distribution on with parameter as defined in [7] has probability density function given by
so the rate constant is . This distribution also has constant rate for the covering graph , with rate constant. .