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Register a new userGreens relations on the deformed transformation semigroups
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Authors
G.Y. Tsyaputa
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Greens relations on the deformed finite inverse symmetric semigroup $mathcal{IS}_n$ and the deformed finite symmetric semigroup $mathcal{T}_n$ are described.
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Pairwise non-isomorphic semigroups obtained from the finite inverse symmetric semigroup $mathcal{IS}_n ,$ finite symmetric semigroup $mathcal{T}_n$ and bicyclic semigroup by the deformed multiplication proposed by Ljapin are classified.
For each subchain $X$ of a chain $X$, let $T_{RE}(X, X)$ denote the semigroup under composition of all full regressive transformations, $alpha:Xrightarrow X$ satisfying $xalphaleq x$ for all $xin X$. Necessary and sufficient conditions for $T_{RE}(X,X)$ and $T_{RE}(Y,Y)$ to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations $T_{RE}(X,X)$ where $X$ are familiar subchains of $R$, the chain of real numbers.
In this paper we characterize those linear bijective maps on the monoid of all $n times n$ square matrices over an anti-negative semifield which preserve and strongly preserve each of Greens equivalence relations $mathcal{L}, mathcal{R}, mathcal{D}, mathcal{J}$ and the corresponding three pre-orderings $leq_mathcal{L}, leq_mathcal{R}, leq_mathcal{J}$. These results apply in particular to the tropical and boolean semirings, and for these two semirings we also obtain corresponding results for the $mathcal{H}$ relation.
First we give a definition of a coverage on a inverse semigroup that is weaker than the one gave by a Lawson and Lenz and that generalizes the definition of a coverage on a semilattice given by Johnstone. Given such a coverage, we prove that there exists a pseudogroup that is universal in the sense that it transforms cover-to-join idempotent-pure maps into idempotent-pure pseudogroup homomorphisms. Then, we show how to go from a nucleus on a pseudogroup to a topological groupoid embedding of the corresponding groupoids. Finally, we apply the results found to study Exels notions of tight filters and tight groupoids.
For a numerical semigroup S $subseteq$ N with embedding dimension e, conductor c and left part L = S $cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $ge$ 0 always holds, a question known since as Wilfs conjecture. Using a closely related lower bound W 0 (S) $le$ W (S), we show that if |L| $le$ 12 then W 0 (S) $ge$ 0, thereby settling Wilfs conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = --1. Wilfs conjecture remains open for |L| $ge$ 13.
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