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The free monoid on a set A is usually denoted A ∗. The free semigroup on A is the subsemigroup of A ∗ containing all elements except the empty string. It is usually denoted A +. [1] [2] More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set. [3]
In addition, corporations may change key aspects of their legal identity, capitalization, or structure in a tax free manner. Examples of reorganizations that may be tax free include mergers, liquidations of subsidiaries, share for share exchanges, exchanges of shares for assets, changes in form or place of organization, and recapitalizations. [60]
The Kleene star is defined for any monoid, not just strings. More precisely, let (M, ⋅) be a monoid, and S ⊆ M. Then S * is the smallest submonoid of M containing S; that is, S * contains the neutral element of M, the set S, and is such that if x,y ∈ S *, then x⋅y ∈ S *.
An S corporation (or S Corp), for United States federal income tax, is a closely held corporation (or, in some cases, a limited liability company (LLC) or a partnership) that makes a valid election to be taxed under Subchapter S of Chapter 1 of the Internal Revenue Code. [1] In general, S corporations do not pay any income taxes.
In essence, the above rule provides that the cooperative corporation need not include this amount paid back to the patrons, as a C corporation ordinarily would. Note that dividends paid out by a cooperative corporation which are not attributable to business done with patrons pursuant to the above definition are still subject to taxation at the ...
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Corporate tax rates generally are the same for differing types of income, yet the US graduated its tax rate system where corporations with lower levels of income pay a lower rate of tax, with rates varying from 15% on the first $50,000 of income to 35% on incomes over $10,000,000, with phase-outs.
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.
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