arXiv subjects

Miloš S. Kurilić

Publications and source records attributed to Miloš S. Kurilić.

27 records · Page 2Linked to original sources

arXiv2018

Back and Forth Systems of Condensations

If $L$ is a relational language, an $L$-structure ${\mathbb X}$ is condensable to an $L$-structure ${\mathbb Y}$, we write ${\mathbb X} \preccurlyeq _c {\mathbb Y}$, iff there is a bijective homomorphism (condensation) from ${\mathbb X}$ onto ${\mathbb Y}$. We characterize the preorder $\preccurlyeq _c$, the corresponding equivalence relation of bi-condensability, ${\mathbb X} \sim _c {\mathbb Y}$, and the reversibility of $L$-structures in terms of back and forth systems and the corresponding games. In a similar way we characterize the ${\mathcal P}_{\infty ω}$-equivalence (which is equivalent to the generic bi-condensability) and the ${\mathcal P}$-elementary equivalence of $L$-structures, obtaining analogues of Karp's theorem and the theorems of Ehrenfeucht and Fraïssé. In addition, we establish a hierarchy between the similarities of structures considered in the paper. Applying these results we show that homogeneous universal posets are not reversible and that a countable $L$-structure ${\mathbb X}$ is weakly reversible (that is, satisfies the Cantor-Schröder-Bernstein property for condensations) iff its ${\mathcal P}_{\infty ω}\cup {\mathcal N}_{\infty ω}$-theory is countably categorical.

math.LO↗

arXiv2018

Reversibility of Extreme Relational Structures

A relational structure $\mathbb{X}$ is called reversible iff each bijective homomorphism from $\mathbb{X}$ onto $\mathbb{X}$ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language $L$ is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language $L$ on a fixed domain $X$ determine reversible structures. We isolate certain syntactical conditions providing that a consistent $L_{\infty ω}$-theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. In particular, we characterize the reversible countable ultrahomogeneous graphs.

math.LO↗

arXiv2017

Reversible Disjoint Unions of Well Orders and Their Inverses

A poset ${\mathbb{P}}$ is called reversible iff every bijective homomorphism $f:{\mathbb{P}} \rightarrow {\mathbb{P}}$ is an automorphism. Let ${\mathcal{W}}$ and ${\mathcal{W}} ^*$ denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the form ${\mathbb{P}} =\bigcup _{i\in I}{\mathbb{L}} _i$, where ${\mathbb{L}} _i, i\in I$, are pairwise disjoint linear orders from ${\mathcal{W}} \cup {\mathcal{W}} ^*$. First, if ${\mathbb{L}} _i \in {\mathcal{W}}$, for all $i\in I$, and ${\mathbb{L}} _i \cong α_i =γ_i+n_i\in Ord$, where $γ_i\in Lim \cup \{0\}$ and $n_i\inω$, defining $I_α:= \{ i\in I : α_i = α\}$, for $α\in Ord$, and $J_γ:= \{ j\in I : γ_j = γ\}$, for $γ\in Lim _0$, we prove that $\bigcup _{i\in I} {\mathbb{L}} _i$ is a reversible poset iff $\langle α_i :i\in I\rangle$ is a finite-to-one sequence, or there is $γ=\max \{ γ_i : i\in I\}$, for $α\leq γ$ we have $|I_α|<ω$, and $\langle n_i : i\in J_γ\setminus I_γ\rangle $ is a reversible sequence of natural numbers. The same holds when ${\mathbb{L}} _i \in {\mathcal{W}} ^*$, for all $i\in I$. In the general case, the reversibility of the whole union is equivalent to the reversibility of the union of components from ${\mathcal{W}}$ and the union of components from ${\mathcal{W}} ^*$.

math.LO↗

arXiv2017

Reversibility of Disconnected Structures

A relational structure is called reversible iff every bijective endomorphism of that structure is an automorphism. We give several equivalents of that property in the class of disconnected binary structures and some its subclasses. For example, roughly speaking and denoting the set of integers by ${\mathbb Z}$, a structure having reversible components is reversible iff its components can not be "merged" by condensations (bijective homomorphisms) and each ${\mathbb Z}$-sequence of condensations between different components must be, in fact, a sequence of isomorphisms. We also give equivalents of reversibility in some special classes of structures. For example, we characterize CSB linear orders of a limit type and show that a disjoint union of such linear orders is a reversible poset iff the corresponding sequence of order types is finite-to-one.

math.LO↗

arXiv2017

Reversible Sequences of Cardinals, Reversible Equivalence Relations, and Similar Structures

A relational structure ${\mathbb X}$ is said to be reversible iff every bijective endomorphism $f:X\rightarrow X$ is an automorphism. We define a sequence of non-zero cardinals $\langle κ_i :i\in I\rangle$ to be reversible iff each surjection $f :I\rightarrow I$ such that $κ_j =\sum_{i\in f^{-1}[\{ j \}]}κ_i$, for all $j\in I $, is a bijection, and characterize such sequences: either $\langle κ_i :i\in I\rangle$ is a finite-to-one sequence, or $κ_i\in {\mathbb N}$, for all $i\in I$, $K:=\{ m\in {\mathbb N} : κ_i =m $, for infinitely many $i\in I \}$ is a non-empty independent set, and $\gcd (K)$ divides at most finitely many elements of the set $\{ κ_i :i\in I \}$. We isolate a class of binary structures such that a structure from the class is reversible iff the sequence of cardinalities of its connectivity components is reversible. In particular, we characterize reversible equivalence relations, reversible posets which are disjoint unions of cardinals $\leq ω$, and some similar structures. In addition, we show that a poset with linearly ordered connectivity components is reversible, if the corresponding sequence of cardinalities is reversible and, using this fact, detect a wide class of examples of reversible posets and topological spaces.

math.LO↗

arXiv2014

Different Similarities

We establish the hierarchy among twelve equivalence relations (similarities) on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language $L$ and a cardinal $κ$, we consider the interplay between the restrictions of these similarities to the class $Mod _L (κ)$ of all $L$-structures of size $κ$. It turns out that, concerning the number of different similarities and the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages (where all these similarities are different).

math.LO↗

arXiv2014

Copies of the Random Graph: the 2-localization

Let $G$ be a countable graph containing a copy of the countable random graph (Erdős-Rényi graph, Rado graph), $Emb (G)$ the monoid of its self-embeddings, ${\mathbb P} (G)=\{f[G]: f\in Emb (G)\}$ the set of copies of $G$ contained in $G$, and ${\mathcal I}_G$ the ideal of subsets of $G$ which do not contain a copy of $G$. We show that the poset $< {\mathbb P} (G), \subset>$, the algebra $P (G)/{\mathcal I}_G$, and the inverse of the right Green's pre-order $< Emb (G),\preceq ^R >$ have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the following law: for each double sequence $[b_{nm}: < n, m > \in ω\times ω]$ of elements of ${\mathbb B}$ $$\textstyle \bigwedge_{n \in ω}\; \bigvee_{m \in ω}\; b_{nm} = \bigvee_{{\mathcal T} \,\in \, Bt ({}^{<ω}ω)}\; \bigwedge_{n \in ω}\; \bigvee_{φ\,\in \,{\mathcal T} \cap {}^{n+1}ω}\; \bigwedge_{k\leq n}\; b_{kφ(k)}, $$ where $Bt ({}^{<ω}ω)$ denotes the set of all binary subtrees of the tree ${}^{<ω}ω$.

math.LO↗

arXiv2014

Copies of the Random Graph

Let $(R, \sim )$ be the Rado graph, $Emb (R)$ the monoid of its self-embeddings, $Π(R)=\{ f[R]: f\in Emb (R)\}$ the set of copies of $R$ contained in $R$, and ${\mathcal I}_R$ the ideal of subsets of $R$ which do not contain a copy of $R$. We consider the poset $( Π(R ), \subset )$, the algebra $P (R)/{\mathcal I _R}$, and the inverse of the right Green's pre-order on $Emb (R)$, and show that these pre-orders are forcing equivalent to a two step iteration of the form $P \ast π$, where the poset $P$ is similar to the Sacks perfect set forcing: adds a generic real, has the $\aleph _0$-covering property and, hence, preserves $ω_1$, has the Sacks property and does not produce splitting reals, while $π$ codes an $ω$-distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph.

math.LO↗

arXiv2013

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

We compare the forcing related properties of a complete Boolean algebra B with the properties of the convergences $λ_s$ (the algebraic convergence) and $λ_{ls}$ on B generalizing the convergence on the Cantor and Aleksandrov cube respectively. In particular we show that $λ_{ls}$ is a topological convergence iff forcing by B does not produce new reals and that $λ_{ls}$ is weakly topological if B satisfies condition $(\hbar)$ (implied by the ${\mathfrak t}$-cc). On the other hand, if $λ_{ls}$ is a weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in some generic extension the distributivity number of the ground model is greater than or equal to the tower number of the extension. So, the statement "The convergence $λ_{ls}$ on the collapsing algebra $B=\ro ((ω_2)^{<ω})$ is weakly topological" is independent of ZFC.

math.GN↗