Remarks on Graphons

The notion of the graphon (a symmetric measurable fuzzy set of $[0, 1]^2$) was introduced by L. Lov\'asz and B. Szegedy in 2006 to describe limit objects of convergent sequences of dense graphs. In their investigation the integral $$t(F,W)=\int _{[0, 1]^k}\prod _{ij\in E(F)}W(x_i,x_j)dx_1dx_2\cdots dx_k$$ plays an important role in which $W$ is a graphon and $E(F)$ denotes the set of all edges of a $k$-labelled simple graph $F$. In our present paper we show that the set of all fuzzy sets of $[0, 1]^2$ is a right regular band with respect to the operation $\circ$ defined by $$(f\circ g)(s,t)=\vee _{(x,y)\in [0, 1]^2}(f(x,y)\wedge g(s,t));\quad (s, t)\in [0, 1]^2,$$ and the set of all graphons is a left ideal of this band. We prove that, if $W$ is an arbitrary graphon and $f$ is a fuzzy set of $[0, 1]^2$, then $$|t(F; W)-t(F; f\circ W)|\leq |E(F)|(\sup(W)-\sup(f))\Delta (\{W> \sup(f)\} )$$ for arbitrary finite simple graphs $F$, where $\Delta (\{W> \sup(f)\})$ denotes the area of the set $\{W>\sup(f)\}$ of all $(x, y)\in [0, 1]^2$ satisfying $W(x,y)>\sup(f)$.Comment: 11 page

Remarks on the paper "M. Kolibiar, On a construction of semigroups"

In his paper "On a construction of semigroups", M. Kolibiar gives a construction for a semigroup $T$ (beginning from a semigroup $S$) which is said to be derived from the semigroup $S$ by a $\theta$-construction. He asserted that every semigroup $T$ can be derived from the factor semigroup $T/\theta (T)$ by a $\theta$-construction, where $\theta (T)$ is the congruence on $T$ defined by: $(a, b)\in \theta (T)$ if and only if $xa=xb$ for all $x\in T$. Unfortunately, the paper contains some incorrect part. In our present paper we give a revision of the paper.Comment: Version v5 differs from the published and the earlier versions in the following: some part of the proof of Theorem 2 are clarified, and some typing errors are correcte

The Algebraic View of Computation

We argue that computation is an abstract algebraic concept, and a computer is a result of a morphism (a structure preserving map) from a finite universal semigroup.Comment: 13 pages, final version will be published elsewher