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To make the tax system more progressive, the Government is raising property tax charges for top-end residential properties in Singapore Finances 2013, with the most important increases making use of to investment properties that are not occupied by their homeowners.
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In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself that is closed under function composition. If it includes the identity function, it is a transformation (or composition) monoid. This is the semigroup anologue of a permutation group.
A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being effective, i.e., if two elements of the semigroup have the same action, then they are equal.
An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set.
Transformation semigroups and monoids
A transformation semigroup is a pair (X,S), where X is a set and S is a semigroup of transformations of X. Here a transformation of X is just a function from X to itself, not necessarily invertible, and therefore S is simply a set of transformations of X which is closed under composition of functions. If S includes the identity transformation of X, then it is called a transformation monoid. Obviously any transformation semigroup S determines a transformation monoid M by taking the union of S with the identity transformation. A transformation monoid whose elements are invertible is a permutation group.
The set of all transformations of X is a transformation monoid called the full transformation monoid (or semigroup) of X. It is also called the symmetric semigroup of X and is denoted by TX. Thus a transformation semigroup (or monoid) is just a subsemigroup (or submonoid) of the full transformation monoid of X. The full transformation monoid is a regular semigroup.
If (X,S) is a transformation semigroup then X can be made into a semigroup action of S by evaluation:
This is a monoid action if S is a transformation monoid.
The characteristic feature of transformation semigroups, as actions, is that they are effective, i.e., if
then s = t. Conversely if a semigroup S acts on a set X by T(s,x) = s • x then we can define, for s ∈ S, a transformation Ts of X by
The map sending s to Ts is injective if and only if (X, T) is effective, in which case the image of this map is a transformation semigroup isomorphic to S.
Cayley representation
In group theory, Cayley's theorem asserts that any group G is isomorphic to a subgroup of the symmetric group of G (regarded as a set), so that G is a permutation group. This theorem generalizes straightforwardly to monoids: any monoid M is a transformation monoid of its underlying set, via the action given by left (or right) multiplication. This action is effective because if ax = bx for all x in M, then by taking x equal to the identity element, we have a = b.
For a semigroup S without a (left or right) identity element, we take X to be the underlying set of the monoid corresponding to S to realise S as a transformation semigroup of X. In particular any finite semigroup can be represented as a subsemigroup of transformations of a set X with |X| ≤ |S| + 1, and if S is a monoid, we have the sharper bound |X| ≤ |S|, as in the case of finite groups.
Transformation monoid of an automaton
Let M be a deterministic automaton with state space S and alphabet A. The words in the free monoid A∗ induce transformations of S giving rise to a monoid morphism from A∗ to the full transformation monoid TS. The image of this morphism is the transformation semigroup of M.[1]
For a regular language, the syntactic monoid is isomorphic to the transformation monoid of the minimal automaton of the language.[2]
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev (2000), Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, Expositions in Mathematics 29, Walter de Gruyter, Berlin, ISBN 978-3-11-015248-7.
- Template:Planetmath reference