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Register a new userCongruences on Menger algebras
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We discuss some types of congruences on Menger algebras of rank $n$, which are generalizations of the principal left and right congruences on semigroups. We also study congruences admitting various types of cancellations and describe their relationship with strong subsets.
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It is a survey of the main results on abstract characterizations of algebras of $n$-place functions obtained in the last 40 years. A special attention is paid to those algebras of $n$-place functions which are strongly connected with groups and semigroups, and to algebras of functions closed with respect natural relations defined on their domains.
Algebraic properties of $n$-place opening operations on a fixed set are described. Conditions under which a Menger algebra of rank $n$ can be represented by $n$-place opening operations are found.
By a completely inverse $AG^{**}$-groupoid we mean an inverse $AG^{**}$-groupoid $A$ satisfying the identity $xx^{-1}=x^{-1}x$, where $x^{-1}$ denotes a unique element of $A$ such that $x=(xx^{-1})x$ and $x^{-1}=(x^{-1}x)x^{-1}.$ We show that the set of all idempotents of such groupoid forms a semilattice and the Greens relations $mathcal{H,L, R,D}$ and $mathcal{J}$ coincide on $A$. The main result of this note says that any completely inverse $AG^{**}$-groupoid meets the famous Lallements Lemma for regular semigroups. Finally, we show that the Greens relation $mathcal{H}$ is both the least semilattice congruence and the maximum idempotent-separating congruence on any completely inverse $AG^{**}$-groupoid.
Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $mathbb{F}$ (assuming $mathrm{char} mathbb{F} e 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $Xin{A,C,D}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r$ with $rle 100$ as well as the asymptotic behaviour of the average, $hat N_X(r)$, for large $r$. In particular, we prove that there exist positive constants $b$ and $c$ such that $exp(br^{2/3})lehat N_X(r)leexp(cr^{2/3})$. The analogous average for matrix algebras $M_n(mathbb{F})$ is proved to be $aln n+O(1)$ where $a$ is an explicit constant depending on $mathrm{char} mathbb{F}$.
We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] leq exp(W)^K$. A G-grading $W = bigoplus_{g in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} eq 0$ for any $r geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.
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