Closure (mathematics)

Summary

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span) or the generated set. Definitions Let S be a set equipped with one or several methods for producing elements of S from other elements of S. A subset X of S is said to be closed under the

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https://en.wikipedia.org/wiki/Closure_(mathematics)

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Expressiveness and Closure Properties for Quantitative Languages

Laurent Doyen

Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages L that assign to each word w a real number L(w). In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-omega-regular for deterministic limit-average and discounted-sum automata, while this set is always omega-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the omega-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages L_1 and L_2, we consider the operations max(L_1,L_2), min(L_1,L_2), and 1-L_1, which generalize the boolean operations on languages, as well as the sum L_1 + L_2. We establish the closure properties of all classes of quantitative languages with respect to these four operations.

2008

Expressiveness And Closure Properties For Quantitative Languages

Laurent Doyen

2010

We derive the closure relation for N polaritons made of three different types of excitons: bosonized excitons, Frenkel, or Wannier excitons. In the case of polaritons made of Wannier excitons, we show how this closure relation, which appears as nondiagonal, may reduce to the one of N elementary bosons, the photons, with its 1/N! prefactor, or to the one of N Wannier excitons, with its (1/N!)(2) prefactor. Widely different forms of closure relations are thus found depending on the composite bosons at hand. Comparison with closure relations of excitons, either bosonized or kept composite as Frenkel or Wannier excitons, allows us to discuss the influence of a reduction in the number of internal degrees of freedom, as well as the importance of the composite nature of the particles and the existence of fermionic components.

2008