288 research outputs found
Monoids admit finite complete rewriting systems
The aim of this note is to prove that monoids , with of relative length 6, admit finite complete
rewriting systems. This is some advance in the understanding the long-standing
open problem whether the word problem for one-relator monoids is soluble
Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure
given by identifying words labelling vertices that appear in the same position
of isomorphic components of the crystal. In the particular case of the crystal
graph for the -analogue of the special linear Lie algebra
, this monoid is the celebrated plactic monoid, whose
elements can be identified with Young tableaux. The crystal graph and the
so-called Kashiwara operators interact beautifully with the combinatorics of
Young tableaux and with the Robinson--Schensted--Knuth correspondence and so
provide powerful combinatorial tools to work with them. This paper constructs
an analogous `quasi-crystal' structure for the hypoplactic monoid, whose
elements can be identified with quasi-ribbon tableaux and whose connection with
the theory of quasi-symmetric functions echoes the connection of the plactic
monoid with the theory of symmetric functions. This quasi-crystal structure and
the associated quasi-Kashiwara operators are shown to interact just as neatly
with the combinatorics of quasi-ribbon tableaux and with the hypoplactic
version of the Robinson--Schensted--Knuth correspondence. A study is then made
of the interaction of the crystal graph of the plactic monoid and the
quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal
structure is applied to prove some new results about the hypoplactic monoid.Comment: 55 pages. Minor revision to fix typos, add references, and discuss an
open questio
Automaton semigroups: new construction results and examples of non-automaton semigroups
This paper studies the class of automaton semigroups from two perspectives:
closure under constructions, and examples of semigroups that are not automaton
semigroups. We prove that (semigroup) free products of finite semigroups always
arise as automaton semigroups, and that the class of automaton monoids is
closed under forming wreath products with finite monoids. We also consider
closure under certain kinds of Rees matrix constructions, strong semilattices,
and small extensions. Finally, we prove that no subsemigroup of arises as an automaton semigroup. (Previously, itself was
the unique example of a finitely generated residually finite semigroup that was
known not to arise as an automaton semigroup.)Comment: 27 pages, 6 figures; substantially revise
Hopfian and co-hopfian subsemigroups and extensions
This paper investigates the preservation of hopficity and co-hopficity on
passing to finite-index subsemigroups and extensions. It was already known that
hopficity is not preserved on passing to finite Rees index subsemigroups, even
in the finitely generated case. We give a stronger example to show that it is
not preserved even in the finitely presented case. It was also known that
hopficity is not preserved in general on passing to finite Rees index
extensions, but that it is preserved in the finitely generated case. We show
that, in contrast, hopficity is not preserved on passing to finite Green index
extensions, even within the class of finitely presented semigroups. Turning to
co-hopficity, we prove that within the class of finitely generated semigroups,
co-hopficity is preserved on passing to finite Rees index extensions, but is
not preserved on passing to finite Rees index subsemigroups, even in the
finitely presented case. Finally, by linking co-hopficity for graphs to
co-hopficity for semigroups, we show that without the hypothesis of finite
generation, co-hopficity is not preserved on passing to finite Rees index
extensions.Comment: 15 pages; 3 figures. Revision to fix minor errors and add summary
table
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