Поговоримо про детерміністичні фрактали, способи їхнього малювання за допомогою гри в хаос, гру в хаос, що керується ДНК-кодом і теж по візуальне розрізнення біологічних видів за допомогою детерміністичних фракталів, згенерованих їх ДНК-кодом.

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У ході семінару буде розглянуто традиційні прийоми біоінформатики, які застосовують для виявлення еволюційної спорідненості (гомології) між генами (нуклеотидними послідовностями) та продуктами їхньої експресії (мРНК, білки), а також цілими геномами.

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We define a new semigroup construction 𝓑^𝓕_ω which generalized the bicyclic minoid and discusse its properties.

We show that 𝓑^𝓕_ω is an inverse semigroup and describe its idempotents.

We describe the Green relations on the semigroup 𝓑^𝓕_ω.

We describe the natural partial order on the semigroup 𝓑^𝓕_ω.

We give the criteria of simplicity, 0-simplicity, bisimplicity, 0-bisimplicity of the semigroup 𝓑^𝓕_ω.

1. 𝓑^𝓕_ω has the identity;
2. 𝓑^𝓕_ω is isomorphic to the bicyclic semigroup; or
3. 𝓑^𝓕_ω is isomorphic to the countable semigroup of matrix units.

Let 𝗄 be any cardinal and 𝓘𝓟𝓕(σ(ℕ^𝗄)) be the semigroup of all order isomorphisms between principal filters of the set σ(ℕ^𝗄) with the product order. We discuss on the group of units of the semigroup 𝓘𝓟𝓕(σ(ℕ^𝗄)).

Let 𝗄 be any cardinal and 𝓘𝓟𝓕(σ(ℕ^𝗄)) be the semigroup of all order isomorphisms between principal filters of the set σ(ℕ^𝗄) with the product order. We discuss on the group of units of the semigroup 𝓘𝓟𝓕(σ(ℕ^𝗄)).

We study the structure of variants (sandwich semigroups) of some Rees matrix semigroups over a group with zero, finite lover semilattices, commutative bands with zero.
The criterion of the isomorphism of two variants is obtained for Rees matrix semigroups over a trivial group with zero, finite lover semilattices, commutative bands with zero, in particular, for the lattice of partition of a countable set. Also we proved that finite Brandt semigroup is not a variant of any semigroup.
Automorphism groups of variants of the power set of a finite set and of the lattice of partition are studied.

We discussed on the results of the paper M.V. Volkov "Semiring identities of the Brandt monoid" arXiv:2103.06077.

A subset $X$ of a group is called decomposable if $X\subseteq XX$ and product-one if there exists a sequence of pairwise distinct elements $x_1,\dots,x_n\in X$ such that $x_1\cdots x_n=1$. We shall discuss a recent result of Lev, Nagy and Pach (2021) who proved that every finite set in an abelian group is product-one. On the other hand, we present an example of a 5-element decomposable set in the Heisenberg group which is not product-one. Also for every finite group $G$ we discuss some lower and upper bound on the largest cardinality $f_1(G)$ of a subset $A\subseteq G$ which is not product-one. We shall prove that $f_1(G)\ge log_2(G)$ for all finite solvable groups, except for the groups $C_2$, $C_3$, $C_3\times C_3$, $D_{10}$, and $(C_3\times C_3):C_2$, which have $f_1(G)$ equal to $1,2,3,3$, and $4$, respectively.

Підмножина $A$ групи $G$ називається сумною, якщо існують попарно різні елементи $a_1,\dots,a_n\in A$ cума яких $a_1+\dots+a_n$ дорівнює нулю. Підмножина групи називається веселою, якщо вона не є сумною. Для скінченної групи $G$ через $f_1(G)$ позначаємо найбільшу потужність веселої множини в групі $G$. Є гіпотеза, що $\lfloor \log_2(|G|)\rfloor\le f_1(G)<\sqrt{2|G|}$ для усіх скінченних груп $G$. Ми доведемо нижню оцінку $\log_2(|G|)\le f_1(G)$ для усіх розв'язних скінченних груп, за винятком груп $C_3,C_5,C_3\times C_3,D_{10},(C_3\times C_3):C_2)$, для яких $f_1(G)$ дорівює $1,2,3,3,4$ i є строго меншим за $\log_2(|G|)$.

Similar as in [O. Gutik, and M. Mykhailenych, On some generalization of the bicyclic monoid, Visn. L'viv. Univ., Ser. Mekh.-Mat. (submitted)], we introduce the algebraic extension $\boldsymbol{B}_{\mathbb{Z}}^{\mathcal{F}}$ of the extended bicyclic semigroup for an arbitrary $\omega$-closed family $\mathcal{F}$ subsets of $\omega$. It is proven that $\boldsymbol{B}_{\mathbb{Z}}^{\mathcal{F}}$ is combinatorial inverse semigroup and Green's relations, the natural partial order on $\boldsymbol{B}_{\mathbb{Z}}^{\mathcal{F}}$ and its set of idempotents are described. We gave the criteria of simplicity, $0$-simplicity, bisimplicity, $0$-bisimplicity of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathcal{F}}$ and when $\boldsymbol{B}_{\mathbb{Z}}^{\mathcal{F}}$ is isomorphic to the extended bicyclic semigroup or the countable semigroup of matrix units. We proved that in the case when the family $\mathcal{F}$ consists of all singletons of $\mathbb{Z}$ and the empty set then the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathcal{F}}$ is isomorphic to the Brandt $\lambda$-extension of the semilattice $(\omega,\min)$.

We study group congruences on the semigroup $\boldsymbol{B}_{\omega}^{\mathcal{F}}$ and its homomorphic retracts in the case when an $\omega$-closed family $\mathcal{F}$ consists of inductive non-empty subsets of $\omega$. It is proven that a congruence $\mathfrak{C}$ on $\boldsymbol{B}_{\omega}^{\mathcal{F}}$ is a group congruence if and only if its restriction on a subsemigroup of $\boldsymbol{B}_{\omega}^{\mathcal{F}}$, which is isomorphic to the bicyclic semigroup, is not the identity relation. Also, all non-trivail homomorphic retracts of the semigroup $\boldsymbol{B}_{\omega}^{\mathcal{F}}$ are described.