214,305 research outputs found
Factorization in Formal Languages
We consider several novel aspects of unique factorization in formal
languages. We reprove the familiar fact that the set uf(L) of words having
unique factorization into elements of L is regular if L is regular, and from
this deduce an quadratic upper and lower bound on the length of the shortest
word not in uf(L). We observe that uf(L) need not be context-free if L is
context-free.
Next, we consider variations on unique factorization. We define a notion of
"semi-unique" factorization, where every factorization has the same number of
terms, and show that, if L is regular or even finite, the set of words having
such a factorization need not be context-free. Finally, we consider additional
variations, such as unique factorization "up to permutation" and "up to
subset"
Generalized U-factorization in Commutative Rings with Zero-Divisors
Recently substantial progress has been made on generalized factorization
techniques in integral domains, in particular -factorization. There has
also been advances made in investigating factorization in commutative rings
with zero-divisors. One approach which has been found to be very successful is
that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work
done in these two areas by generalizing -factorization to rings with
zero-divisors by using the notion of U-factorization.Comment: 16 pages, to appear in Rocky Mountain Journal of Mathematic
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