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Register a new userFree left and right adequate semigroups
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Authors
Mark Kambites
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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.
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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.
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 $left( A,Bright)inmathfrak{sl}left( n,mathbb{H}right)^{2}$.
The concept of a k-translatable groupoid is explored in depth. Some properties of idempotent k-translatable groupoids, left cancellative k-translatable groupoids and left unitary k-translatable groupoids are proved. Necessary and sufficient conditions are found for a left cancellative k-translatable groupoid to be a semigroup. Any such semigroup is proved to be left unitary and a union of disjoint copies of cyclic groups of the same order. Methods of constructing k-translatable semigroups that are not left cancellative are given.
We construct the minimal supersymmetric left-right theory and show that at the renormalizable level it requires the existence of an intermediate $B-L$ breaking scale. The subsequent symmetry breaking down to MSSM automatically preserves R-symmetry. Furthermore, unlike in the nonsupersymmetric version of the theory, the see-saw mechanism takes its canonical form. The theory predicts the existence of a triplet of Higgs scalars much lighter than the $B-L$ breaking scale.
In this paper, we investigate *-DMP elements in $*$-semigroups and $*$-rings. The notion of *-DMP element was introduced by Patr{i}cio in 2004. An element $a$ is *-DMP if there exists a positive integer $m$ such that $a^{m}$ is EP. We first characterize *-DMP elements in terms of the {1,3}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we give the pseudo core decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for matrices to an arbitrary $*$-ring; and this decomposition turns to be a useful tool to characterize *-DMP elements. Further, we extend Wangs core-EP order from matrices to $*$-rings and use it to investigate *-DMP elements. Finally, we give necessary and sufficient conditions for two elements $a,~b$ in $*$-rings to have $aa^{scriptsizetextcircled{tiny D}}=bb^{scriptsizetextcircled{tiny D}}$, which contribute to investigate *-DMP elements.
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