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Page "Monoid" ¶ 15
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Every and singleton
Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms.
* Every singleton set

Every and set
: Every set has a choice function.
Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.
** Well-ordering theorem: Every set can be well-ordered.
** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
** Antichain principle: Every partially ordered set has a maximal antichain.
* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).
: Every non-empty set A contains an element B which is disjoint from A.
* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.
Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.
* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.
Every corporation, whether financial or union, as well as every division of the administration, were set up as branches of the party, the CEOs, Union leaders, and division directors being sworn-in as section presidents of the party.
Every DNS zone must be assigned a set of authoritative name servers that are installed in NS records in the parent zone, and should be installed ( to be authoritative records ) as self-referential NS records on the authoritative name servers.
Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.
# " Personality " Argument: this argument is based on a quote from Hegel: " Every man has the right to turn his will upon a thing or make the thing an object of his will, that is to say, to set aside the mere thing and recreate it as his own ".
Every atom across this plane has an individual set of emission cones .</ p > < p > Drawing the billions of overlapping cones is impossible, so this is a simplified diagram showing the extents of all the emission cones combined.
Every processor or processor family has its own machine code instruction set.
Every set is a class, no matter which foundation is chosen.
Every non-empty totally ordered set is directed.
* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

singleton and set
* Any singleton set
The notation is used, because it is the factor ring of by the ideal containing all integers divisible by n, where is the singleton set.
An event, however, is any subset of the sample space, including any singleton set ( an elementary event ), the empty set ( an impossible event, with probability zero ) and the sample space itself ( a certain event, with probability one ).
If the negated query can be refuted, i. e., an instantiation for all free variables is found that makes the union of clauses and the singleton set consisting of the negated query false, it follows that the original query, with the found instantiation applied, is a logical consequence of the program.
:* is a singleton set and
Since every singleton set has one-dimensional Lebesgue measure zero,
In all abelian groups every conjugacy class is a set containing one element ( singleton set ).
In particular, each singleton is an open set in the discrete topology.
* The empty set is the unique initial object in the category of sets ; every one-element set ( singleton ) is a terminal object in this category ; there are no zero objects.
The singletons are not initial: while every non-empty set admits a function from a singleton, this function is in general not unique.
* Points are closed in X ; i. e. given any x in X, the singleton set
that is, the singleton set containing the empty tuple.
For example, in the category of sets the categorical product is the usual Cartesian product, and the terminal object is a singleton set.
* trivial ring: a ring defined on a singleton set.
MakeSet ( u ) removes u to a singleton set, Find ( u ) returns the standard representative of the set containing u, and Union ( u, v ) merges the set containing u with the set containing v.
In words, there is a set I ( the set which is postulated to be infinite ), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton

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