(adsbygoogle = window.adsbygoogle || []).push({}); Monoid of "specifications" for a group

The question of whether there is any standard math associated with specifications of ordered pairs on a group went nowhere (https://www.physicsforums.com/showthread.php?t=640395), so I will spell out what I have in mind.

It appears possible to define a monoid of "specifications" for a group G, as sketched below. Is there a technical name for this monoid? Is it a special case of some standard structure in group theory?

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Let G be a group. Each element [itex] g \in [/itex] G can be identified with the function that permutes th-e elements of [itex] G [/itex] by left multiplication. In what follows we will consider [itex] g [/itex] to be the set of ordered pairs [itex] \{(x,gx): x \in G \} [/itex].

Define aspecification[itex] s [/itex] on [itex] G [/itex] to be a set of ordered pairs of elements of [itex] G [/itex] such that there exists an element [itex] g \in G [/itex] such that [itex] s \subset g [/itex].

For example, Let [itex] G [/itex] be the group of permutations on the set of 4 letters [itex] \{a,b,c,d\} [/itex] then the set [itex] s = \{(a,b),(c,a)\} [/itex] is a specification since the group element [itex] g_1 = \{(a,b),(b, c),(c,a),(d,d) \} [/itex] contains [itex] s [/itex] as a subset. (There is also another group element [itex] g_2 = \{(a,b),(b,d),(c,a),(d,c)\} [/itex] that contains [itex] s [/itex].) In general, a specification need not define a unique group element.)

For a specification [itex] s [/itex] on a group [itex] G [/itex] , denote by [itex] G(s) [/itex] the set of elements in [itex] G [/itex] that contain [itex] s [/itex].

Examples:

In the previous example [itex] G(s) = \{g_1,g_2\} [/itex].

If [itex] g \in G [/itex] then [itex] G(g) = g [/itex] since the ordered pairs of [itex] g [/itex] define it uniquely.

[itex] G\{\emptyset \} [/itex] is the entire set of elements of [itex] G [/itex].

Define a multiplication operation on two specifications as follows:

Let [itex] s,t [/itex] be specifications on the group [itex] G [/itex]. Define the product [itex] s t [/itex] to be the specification consisting the all ordered pairs [itex] {x,y} [/itex] such there is some ordered pair [itex] (a,y) \in s [/itex] and some ordered pair [itex] (x,a) \in t [/itex]

A specification defines a 1-1 function from a subset of [itex] G [/itex] onto another subset of [itex] G [/itex]. The product of two specifications amounts taking the composition of two such functions on the intersection of their domains.

The identity element [itex] I [/itex] of the group [itex] G [/itex] defines a specification that is a multiplicative identity for the above product operation. The set of all possible specifications for a group [itex] G [/itex] forms a monoid under the product operation.

The monoid of specifications is not the same as a monoid formed by subsets of the group, i.e., in general, [itex] G(s t) [/itex] need not equal [itex] G(s) G(t) [/itex].

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# Monoid of specifications for a group

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