Structure of a permutable Munn semigroup of finite rank

A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 742–746, June, 2006.     

The rank of the inverse semigroup of partial automorphisms on a finite fence

Semigroup Forum - A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $$\mathscr {FI}_{n}$$ of all...     

Structure of a finite inverse semigroup with zero every stable order on which is fundamental or antifundamental

We establish necessary and sufficient conditions for any stable order on a finite inverse semigroup with zero to be fundamental or antifundamental.     

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

LetCom _t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx ^t =x ^t+q . The varietyCom _1,1 is the variety of finite semilattices also denoted byJ ₁. In this paper, we consider the product varietyJ ₁*Com _t,q generated by all semidirect products of the formM*N withM∈J ₁ andN∈Com _t,q . We give a complete sequence of equations forJ ₁*Com _t,q implying complete sequences of equations forJ ₁*(Com∩A),J ₁*(Com∩G) andJ ₁*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups. This material is based upon work supported by the National Science Foundation under Grants No. CCR-9101800 and CCR-9300738. Many thanks to the referee for his valuable comments and suggestions.     

Burnside algebra of a finite inverse semigroup

The Burnside algebra for a finite inverse semigroup over a field is considered (the analog of the Grothendieck algebra). The conditions for the algebra to be Frobenius are investigated. It is shown that, if all the subgroups in the semigroup are commutative, then its Burnside algebra is Frobenius if and only if the order of any maximal subgroup is not divisible by the square of the field characteristic.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 46, pp. 41–52, 1974.     

Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero

We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1353–1362, October, 2007.     

Semisimplicity of the algebra of indecomposable representations of a finite commutative semigroup

Finite semigroups S, having only a finite number of indecomposable matrix representations over a field K of characteristic are considered. It is proved that the algebra over the field F of K-representations of the semigroup S is semisimple if and only if for any maximal subgroup of the semigroup S the characteristic of the field F does not divide the index of its Sylow p-subgroup and is greater than the order of this Sylow subgroup.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 125–128, 1976.     

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 52–60, January, 2009.     

On nilpotent subsemigroups of a finite symmetrical inverse semigroup

This work was supported by the State Foundation for Fundamental Research of SCST of the Ukrain.     

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or ``mirror inner". In particular, we answer a question of B. I. Plotkin.

     

Center of the semigroup algebra of a finite inverse semigroup over the field of complex numbers

In the semigroup algebra A of a finite inverse semigroup S over the field of complex numbers to an indempotent e there is assigned the sum (e)=e+(–1)^Ke_L1e_iK, where e_i,...,e_m are maximal preidempotents of the idempotent e, and the summation goes over all nonempty subsets {i₁,...,i_k} of the set {1,...m} Then for any class K of conjugate group elements of the semigroup S the element K=a·(a^–1a) (the summation goes over all ag) is a central element of the algebra A, and the set {K} of all possible such elements is a basis for the center of the algebra A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 154–158, 1978.     

On the graph for which there is a tree as the inverse interchange graph of a local graph

Let G be a finite, connected, undirected graph without loops and multiple edges. The note modifies slightly the concept of I^–1 (T_t), the inverse interchange graph of the local graph G(T_t) defined by a reference tree t G, and considers the properties of the graph G, when I^–1(T_t) is a tree.     

Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

We prove that the minimal cardinality of a semitransitive subsemigroup in the singular part $\mathcal{I}_{n}\setminus \mathcal{S}_{n}$ of the symmetric inverse semigroup $\mathcal{I}_{n}$ is 2n?p+1, where p is the greatest proper divisor of n, and classify all semitransitive subsemigroups of this minimal cardinality.     

Correction to the paper “semisimplicity of the algebra of indecomposable representations of a finite commutative semigroup”

The result indicated in the title of this paper (Ref. Zh. Mat. 1976, 7A225) is false. The error is corrected.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 112–113, 1979.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.