2019
Том 71
№ 11

# Derech V. D.

Articles: 15
Article (Ukrainian)

### Finite structurally uniform groups and commutative nilsemigroups

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.

Brief Communications (Ukrainian)

### 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.

Article (Ukrainian)

### 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 $\rho \circ \sigma = \sigma \circ \rho$.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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., ξφ = φξ 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.

Article (Ukrainian)

### 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.