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Register a new userTransformation Semigroups with the Deformed Multiplication
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G.Y. Tsyaputa
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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.
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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.
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.
We transform the oscillator algebra with kappa-deformed multiplication rule, proposed in [1],[2], into the oscillator algebra with kappa-deformed flip operator and standard multiplication. We recall that the kappa-multiplication of the kappa-oscillators puts them off-shell. We study the explicit forms of modified mass-shell conditions in both formulations: with kappa-multiplication and with kappa-flip operation. On the example of kappa-deformed 2-particle states we study the clustered nonfactorizable form of the kappa-deformed multiparticle states. We argue that the kappa-deformed star product of two free fields leads in similar way to a nonfactorizable kappa-deformed bilocal field. We conclude with general remarks concerning the kappa-deformed n-particle clusters and kappa-deformed star product of n fields.
The multiplicative semigroup $M_n(F)$ of $ntimes n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our guiding examples of such $S$ are the 2-element Boolean semiring $mathbb{B}$ and the tropical semiring $mathbb{T}$. In the first case, $M_n(mathbb{B})$ is isomorphic to the semigroup of binary relations on an $n$-element set, and in the second, $M_n(mathbb{T})$ is the semigroup of $ntimes n$ tropical matrices. Ilin has proved that for any semiring $R$ and $n>2$, the semigroup $M_n(R)$ is regular if and only if $R$ is a regular ring. We therefore base our investigations for $M_n(S)$ and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of $M_n(S)$, including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.
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.
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