1,511 research outputs found
New polynomial and multidimensional extensions of classical partition results
In the 1970s Deuber introduced the notion of -sets in
and showed that these sets are partition regular and contain all linear
partition regular configurations in . In this paper we obtain
enhancements and extensions of classical results on -sets in two
directions. First, we show, with the help of ultrafilter techniques, that
Deuber's results extend to polynomial configurations in abelian groups. In
particular, we obtain new partition regular polynomial configurations in
. Second, we give two proofs of a generalization of Deuber's
results to general commutative semigroups. We also obtain a polynomial version
of the central sets theorem of Furstenberg, extend the theory of
-systems of Deuber, Hindman and Lefmann and generalize a classical
theorem of Rado regarding partition regularity of linear systems of equations
over to commutative semigroups.Comment: Some typos, including a terminology confusion involving the words
`clique' and `shape', were fixe
Recognizing pro-R closures of regular languages
Given a regular language L, we effectively construct a unary semigroup that
recognizes the topological closure of L in the free unary semigroup relative to
the variety of unary semigroups generated by the pseudovariety R of all finite
R-trivial semigroups. In particular, we obtain a new effective solution of the
separation problem of regular languages by R-languages
A minimal nonfinitely based semigroup whose variety is polynomially recognizable
We exhibit a 6-element semigroup that has no finite identity basis but
nevertheless generates a variety whose finite membership problem admits a
polynomial algorithm.Comment: 16 pages, 3 figure
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