### A few Remarks on Reduced Ideal-products

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By a chordal graph is meant a graph with no induced cycle of length $\ge 4$. By a ternary system is meant an ordered pair $(W,T)$, where $W$ is a finite nonempty set, and $T\subseteq W\times W\times W$. Ternary systems satisfying certain axioms (A1)–(A5) are studied in this paper; note that these axioms can be formulated in a language of the first-order logic. For every finite nonempty set $W$, a bijective mapping from the set of all connected chordal graphs $G$ with $V\left(G\right)=W$ onto the set of all ternary systems $(W,T)$ satisfying the axioms (A1)–(A5) is...

We study the extensibility of piecewise polynomial functions defined on closed subsets of ${\mathbb{R}}^{2}$ to all of ${\mathbb{R}}^{2}$. The compact subsets of ${\mathbb{R}}^{2}$ on which every piecewise polynomial function is extensible to ${\mathbb{R}}^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $\mathbb{R}$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial...

We extend a result of M. Tamm as follows:Let $f:A\to \mathbb{R},\phantom{\rule{0.166667em}{0ex}}A\subseteq {\mathbb{R}}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto {x}^{r}:(0,\infty )\to \mathbb{R},\phantom{\rule{0.166667em}{0ex}}r\in \mathbb{R}$. Then there exists $N\in \mathbb{N}$ such that for all $(a,b)\in A$, if $y\mapsto f(a,y)$ is ${C}^{N}$ in a neighborhood of $b$, then $y\mapsto f(a,y)$ is real analytic in a neighborhood of $b$.

We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...

We present an abstract equational framework for the specification of systems having both observational and computational features. Our approach is based on a clear separation between the two categories of features, and uses algebra, respectively coalgebra to formalise them. This yields a coalgebraically-defined notion of observational indistinguishability, as well as an algebraically-defined notion of reachability under computations. The relationship between the computations yielding new system states...

We present an abstract equational framework for the specification of systems having both observational and computational features. Our approach is based on a clear separation between the two categories of features, and uses algebra, respectively coalgebra to formalise them. This yields a coalgebraically-defined notion of observational indistinguishability, as well as an algebraically-defined notion of reachability under computations. The relationship between the computations yielding new system...