# Derech V. D.

### Finite structurally uniform groups and commutative nilsemigroups

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1072-1084

Let $S$ be a finite semigroup. By $\mathrm{S}\mathrm{u}\mathrm{b}(S)$ we denote the lattice of all its subsemigroups. If $A \in \mathrm{S}\mathrm{u}\mathrm{b}(S)$, then by $h(A)$ we denote the height of the subsemigroup $A$ in the lattice $\mathrm{S}\mathrm{u}\mathrm{b}(S)$. A semigroup $S$ is called structurally uniform if, for any $A, B \in \mathrm{S}\mathrm{u}\mathrm{b}(S)$ the condition $h(A) = h(B) implies that A \sim = B$. We present a classification of finite structurally uniform groups and commutative nilsemigroups.

### Complete classification of finite semigroups for which the inverse monoid of local automorphisms is a permutable semigroup

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1571-1578

A semigroup $S$ is called permutable if $\rho \circ \sigma = \sigma \circ \rho$. for any pair of congruences $\rho, \sigma$ on $S$. A local automorphism of semigroup $S$ is defined as an isomorphism between two of its subsemigroups. The set of all local automorphisms of the semigroup $S$ with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a complete classification of finite semigroups for which the inverse monoid of local automorphisms is permutable.

### Classification of finite nilsemigroups for which the inverse monoid of local automorphisms is permutable semigroup

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 610-624

A semigroup $S$ is called permutable if $\rho \circ \sigma = \sigma \circ \rho$ for any pair of congruences $\rho$, $\sigma$ on $S$. A local automorphism of the semigroup $S$ is defined as an isomorphism between two subsemigroups of this semigroup. The set of all local automorphisms of a semigroup $S$ with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. In the proposed paper, we present a classification of all finite nilsemigroups for which the inverse monoid of local automorphisms is permutable. Полугруппа $S$ называется перестановочной, если для любой пары конгруэнций $\rho$, $\sigma$ на $S$ имеет место равенство $\rho \circ \sigma = \sigma \circ \rho$.

### Classification of Finite Commutative Semigroups for Which the Inverse Monoid of Local Automorphisms is a ∆-Semigroup

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 867-873

A semigroup $S$ is called a ∆-semigroup if the lattice of its congruences forms a chain relative to the inclusion. A local automorphism of the semigroup $S$> is called an isomorphism between its two subsemigroups. The set of all local automorphisms of the semigroup $S$ relative to the ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is a ∆-semigroup.

### Stable Quasiorderings on Some Permutable Inverse Monoids

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 445–457

Let *G* be an arbitrary group of bijections on a finite set. By *I*(*G*), we denote the set of all injections each of which is included in a bijection from *G*. The set *I*(*G*) forms an inverse monoid with respect to the ordinary operation of composition of binary relations. We study different properties of the semi-group *I*(*G*). In particular, we establish necessary and sufficient conditions for the inverse monoid *I*(*G*) to be permutable (i.e., *ξ* ○ *φ* = *φ* ○ *ξ* for any pair of congruences on *I*(*G*)). In this case, we describe the structure of each congruence on *I*(*G*). We also describe the stable orderings on *I*(*A* _{ n }), where *A* _{ n } is an alternating group.

### On One Class of Factorizable Fundamental Inverse Monoids

Ukr. Mat. Zh. - 2013. - 65, № 6. - pp. 780–786

Let *G* be an arbitrary group of bijections on a finite set and let *I*(*G*) denote the set of all partial injective transformations each of which is included in a bijection from *G*. The set *I*(*G*) is a fundamental factorizable inverse semigroup. We study various properties of the semigroup *I*(*G*). In particular, we describe the automorphisms of *I*(*G*) and obtain necessary and sufficient conditions for each stable order on *I*(*G*) to be fundamental or antifundamental.

### Classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 176-184

We give a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable.

### Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1218-1226

For a semigroup $S$, the set of all isomorphisms between subsemigroups of $S$ is an inverse monoid with respect to composition, which is denoted by $P A(S)$ and is called the monoid of local automorphisms of $S$. A semigroup $S$ is called permutable if, for any pair of congruences $p, \sigma$ on $S$, one has $p \circ \sigma = \sigma \circ p$. We describe the structure of a finite commutative inverse semigroup and a finite band whose monoids of local automorphisms are permutable.

### Structure of finite inverse semigroup with zero, in which every stable order is fundamental or antifundamental

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 29 - 39

We find necessary and sufficient conditions for any stable order on a finite inverse semigroup with zéro to be fondamental or antifundamental.

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

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 52-60

We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental.

### On maximal stable orders on an inverse semigroup of finite rank with zero

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1035–1041

We consider maximal stable orders on semigroups that belong to a certain class of inverse semigroups of finite rank.

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

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1353–1362

We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero.

### Structure of a permutable Munn semigroup of finite rank

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 742–746

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.

### Congruences of a Permutable Inverse Semigroup of Finite Rank

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 469–473

We describe the structure of any congruence of a permutable inverse semigroup of finite rank.

### On Permutable Congruences on Antigroups of Finite Rank

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 346-351

We find necessary and sufficient conditions for any two congruences on an antigroup of finite rank to be permutable.