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Register a new userRight, left and double division in semigroups that are semilattices of groups
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Authors
R. A. R. Monzo
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The binary products of right, left or double division in semigroups that are semilattices of groups give interesting groupoid structures that are in one to one correspondence with semigroups that are semilattices of groups. This work is inspired by the known one to one correspondence between groups and Ward quasigroups.
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Recent research of the author has given an explicit geometric description of free (two-sided) adequate semigroups and monoids, as sets of labelled directed trees under a natural combinatorial multiplication. In this paper we show that there are natural embeddings of each free right adequate and free left adequate semigroup or monoid into the corresponding free adequate semigroup or monoid. The corresponding classes of trees are easily described and the resulting geometric representation of free left adequate and free right adequate semigroups is even easier to understand than that in the two-sided case. We use it to establish some basic structural properties of free left and right adequate semigroups and monoids.
We consider the lattice of subsemigroups of the general linear group over an Artinian ring containing the group of diagonal matrices and show that every such semigroup is actually a group.
Let $mathrm{Sl}left( n,mathbb{H}right)$ be the Lie group of $ntimes n$ quaternionic matrices $g$ with $leftvert det grightvert =1$. We prove that a subsemigroup $S subset mathrm{Sl}left( n,mathbb{H}right)$ with nonempty interior is equal to $mathrm{Sl}left( n,mathbb{H}right)$ if $S$ contains a subgroup isomorphic to $mathrm{Sl}left( 2,mathbb{H}right)$. As application we give sufficient conditions on $A,Bin mathfrak{sl}left( n,mathbb{H}right)$ to ensuring that the invariant control system $dot{g}=Ag+uBg$ is controllable on $mathrm{Sl}left( n,mathbb{H}right)$. We prove also that these conditions are generic in the sense that we obtain an open and dense set of controllable pairs $left( A,Bright)inmathfrak{sl}left( n,mathbb{H}right)^{2}$.
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Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.
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