We discuss on the results of the paper Igor Dolinka and James East, Semigroups of rectangular matrices under a sandwich operation, Semigroup Forum (2018) 96: 253–300. https://doi.org/10.1007/s00233-017-9873-6

We discuss on the results of the paper Emil Daniel Schwab, A half-factorial locally right Garside monoid and the inverse monoid of cofinite monotone partial bijections on $\mathbb{N}^*$, Semigroup Forum (2016) 92: 393–413. https://doi.org/10.1007/s00233-015-9727-z

We discuss on the results of the paper Emil Daniel Schwab, A half-factorial locally right Garside monoid and the inverse monoid of cofinite monotone partial bijections on $\mathbb{N}^*$, Semigroup Forum (2016) 92: 393–413. https://doi.org/10.1007/s00233-015-9727-z

We discuss on the results of the paper Emil Daniel Schwab, A half-factorial locally right Garside monoid and the inverse monoid of cofinite monotone partial bijections on $\mathbb{N}^*$, Semigroup Forum (2016) 92: 393–413. https://doi.org/10.1007/s00233-015-9727-z

We discuss on the results of the paper Emil Daniel Schwab, A half-factorial locally right Garside monoid and the inverse monoid of cofinite monotone partial bijections on $\mathbb{N}^*$, Semigroup Forum (2016) 92: 393–413. https://doi.org/10.1007/s00233-015-9727-z

A Hausdorff topology $\tau$ on the bicyclic monoid $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We introduce a notion of a shift-stable filter on $\omega$ and show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice $\mathcal{SS}^1{\times}\mathcal{SS}^1$ where $\mathcal{SS}^1$ is the set of all shift-stable filters on $\omega$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.
This is joint work with O. Gutik.

A Hausdorff topology $\tau$ on the bicyclic monoid $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We introduce a notion of a shift-stable filter on $\omega$ and show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice $\mathcal{SS}^1{\times}\mathcal{SS}^1$ where $\mathcal{SS}^1$ is the set of all shift-stable filters on $\omega$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.
This is joint work with O. Gutik.

We discuss on the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of psitive integers $\mathbb{N}$ with the usual metric. We describe the elements of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ as partial transformation of $\mathbb{N}^n$, the group of units $H(\mathbb{I})$ and subsets of idempotents of $\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's relations on $\mathbf{I}\mathbb{N}_{\infty}^n$.
This is joint work with A. Savchuk.

We discuss on the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of psitive integers $\mathbb{N}$ with the usual metric. We describe the group of units $H(\mathbb{I})$ and subsets of idempotents of $\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's relations on $\mathbf{I}\mathbb{N}_{\infty}^n$. In particular we show that the quotient semigroup $\mathbf{I}\mathbb{N}_{\infty}^n/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence on $\mathbf{I}\mathbb{N}_{\infty}^n$, is isomorphic to the symmetric group $\mathcal{S}_n$ and $\mathcal{D}=\mathcal{J}$ on $\mathbf{I}\mathbb{N}_{\infty}^n$.
This is joint work with A. Savchuk.

We discuss on the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of psitive integers $\mathbb{N}$ with the usual metric. We prove that for $n\geqslant 2$ the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$ is isomorphic to the semidirect product ${\mathcal{S}_n\ltimes_\mathfrak{h}(\mathcal{P}_{\infty}(\mathbb{N}^n),\cup)}$ of free semilattice with the unit $(\mathcal{P}_{\infty}(\mathbb{N}^n),\cup)$ by the symmetric group $\mathcal{S}_n$.
This is joint work with A. Savchuk.

We discuss on the results of the paper Robert Jajcay, Tatiana Jajcayova, Nóra Szakács, Mária B. Szendrei, Inverse monoids of partial graph automorphisms, arXiv:1809.04422.