The Finite Basis Problem for Kiselman Monoids

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke--Kiselman monoids including the Kiselman monoids $\mathcal{K}_n$. As a consequence, we conclude that the identities of $\mathcal{K}_n$ are nonfinitely based for every $n\ge 4$ and exhibit a finite identity basis for the identities of each of the monoids $\mathcal{K}_2$ and $\mathcal{K}_3$. In the third version a question left open in the initial submission has beed answered.Comment: 16 pages, 1 table, 1 figur

From AA to BB to ZZ

The variety generated by the Brandt semigroup ${\bf B}_2$ can be defined within the variety generated by the semigroup ${\bf A}_2$ by the single identity $x^2y^2\approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${\bf B}_2^1$ and ${\bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${\bf A}_2^1$ that satisfy $x^2y^2\approx y^2x^2$ and contain ${\bf B}_2^1$. A further consequence is that the variety of ${\bf B}_2^1$ cannot be defined within the variety of ${\bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${\bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2y\approx yx^2$ and containing the monoid $M({\bf z}_\infty)$, where ${\bf z}_\infty$ denotes the infinite limit of the Zimin words ${\bf z}_0=x_0$, ${\bf z}_{n+1}={\bf z}_n x_{n+1}{\bf z}_n$

Construction of spirocarbocycles via gold-catalyzed intramolecular dearomatization of naphthols.

A highly efficient, gold-catalyzed intramolecular dearomatization reaction of naphthols via 5-endo-dig cyclization is described. This facile and direct approach furnishes spirocarbocycles in excellent yields under mild conditions

An IT Professional Talents Training Model in Colleges Based on Animal Cell Structure

Under the current period background of big data and cloud computing, there is a huge demand for professionals in related fields such as information technology (IT). To solve this problem, this paper puts forward an IT professional talents training model based on animal cell structure by comparing the structures of animal cells and its efficient operation principle with IT professional training model system. According to the efficient-working principle of ‘Nucleus-Cytoplasm- Environment’, this model is built as a ‘Class (The Core)-College (Internal Environment)-Enterprise (External Environment)’ training model for IT-majored students. The motivation is to cultivate students’ abilities in these four aspects: structure, application, analysis and innovation, namely, regarding theory teaching as the core, college practice training as the pulling force and enterprise project resources as the pushing force. The reliability and validation of this model have been demonstrated by simulation results in Wuhan University of Science and Technology

Finite Basis Problem for Semigroups of Order Six

Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist 15973 pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other 15969 distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order