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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]
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 *.
A set S equipped with a binary operation S × S → S, which we will denote •, is a monoid if it satisfies the following two axioms: Associativity For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds. Identity element There exists an element e in S such that for every element a in S, the equalities e • a = a and a ...
That's enough to cover its high-yielding dividend (over 4% at the current share price) by about 2.2 times. Williams retains the rest of its cash flow to invest in expansion projects and strengthen ...
Let be an alphabet: the set of words over is a monoid, the free monoid on . The recognizable subsets of A ∗ {\displaystyle A^{*}} are precisely the regular languages . Indeed, such a language is recognized by the transition monoid of any automaton that recognizes the language.
A Lyndon word over a totally ordered alphabet A is a word that is lexicographically less than all its rotations. [1] The Chen–Fox–Lyndon theorem states that every string may be formed in a unique way by concatenating a lexicographically non-increasing sequence of Lyndon words.
In the case of a Roth IRA or Roth 401(k), those dividends can be 100% tax-free. Tax-loss harvesting is an additional strategy that can help reduce your dividend taxes.
From a revolutionary restaurant in the 1940s to the world's largest fast-food chain today, McDonald's Corporation (NYSE: MCD) is a staple of American culture. The company has over 41,000 locations ...