arXiv subjects

Miloš S. Kurilić

Publications and source records attributed to Miloš S. Kurilić.

At least 19 recordsLinked to original sources

arXiv2026

Vaught's Conjecture for Sums of Products of Rooted Trees

We confirm Vaught's conjecture for each partial order ${\mathbb X}=\sum _{\mathbb I}\prod _{j 1$ is first-order definable; (f) Each $ω$-categorical partial order from the class $\langle \langle {\mathcal C}^{\rm rt}_{\rm fb}\rangle _Π\rangle _Σ$, where ${\mathcal C}^{\rm rt}_{\rm fb}$ is the class of finite-branching rooted trees, is finitely axiomatizable. Statement (f) is related to the results of Rosenstein (for the class of linear orders) and Schmerl (for the class of partial orders of finite width).

math.LO↗

arXiv2026

Vaught's Conjecture for Unions of Products of Rooted Trees

Let ${\mathcal C} ^{\rm rt}$ be the class of rooted trees and $\langle {\mathcal C} ^{\rm rt}\rangle _{\dot{\cup }Π}$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\mathbb X}= \dot{\bigcup} _{i<n}\prod _{j<m_i}{\mathbb X}_i^j$, where ${\mathbb X}_i^j$ are rooted trees. Defining ${\mathcal T}=\mathop{\rm Th} ({\mathbb X})$, ${\mathcal T} _i ^j=\mathop{\rm Th}({\mathbb X}_i^j)$, for $i<n$ and $j<m_i$, and $κ= \prod _{i<n}\prod _{j<m_i}I({\mathcal T} _i^j)$, we have (a) Vaught's conjecture is true for ${\mathcal T}$: $I({\mathcal T})=κ$, if $κ\in \{ 1,ω,{\mathfrak{c}}\}$, and, otherwise, $I({\mathcal T}) \in [3,ω)$; (b) ${\mathbb Y} \equiv {\mathbb X}$ iff $\;{\mathbb Y} \cong \dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb Y} _i^j$, where ${\mathbb Y}_i^j\equiv {\mathbb X}_i^j$, for $i<n$ and $j<m_i$; (c) ${\mathbb E}\preccurlyeq {\mathbb X}$ iff $\;{\mathbb E} =\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb E}_i^j$, where ${\mathbb E}_i^j\preccurlyeq {\mathbb X}_i^j$, for $i<n$ and $j<m_i$; (d) ${\mathcal T}$ is atomic iff $\;{\mathcal T} _i^j$, for $i<n$ and $j<m_i$, are atomic; then $\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb A}_i^j$ is a countable atomic model of ${\mathcal T}$, where ${\mathbb A}_i^j$ is a countable atomic model of ${\mathcal T} _i^j$, for $i<n$ and $j<m_i$; (e) ${\mathcal T}$ is small iff $\;{\mathcal T} _i^j$, for $i<n$ and $j<m_i$, are small; then $\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb S}_i^j$ is a countably saturated model of ${\mathcal T}$, where ${\mathbb S}_i^j$ is a countably saturated model of ${\mathcal T}_i^j$, for $i<n$ and $j<m_i$.

math.LO↗

arXiv2026

Vaught's Conjecture and Theories of Partial Order Admitting a Finite Lexicographic Decomposition

A complete theory ${\mathcal T}$ of partial order is an FLD$_1$-theory iff some (equivalently, any) of its models ${\mathbb X}$ admits a finite lexicographic decomposition ${\mathbb X} =\sum _{\mathbb I}{\mathbb X} _i$, where ${\mathbb I}$ is a finite partial order and ${\mathbb X} _i$-s are partial orders with a largest element. Then we write $\sum _{\mathbb I}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ and call $\sum _{\mathbb I}{\mathbb X}_i$ a VC-decomposition (resp. a VC$^\sharp$-decomposition} iff ${\mathbb X} _i$ satisfies Vaught's conjecture (VC) (resp. VC$^\sharp$: $I({\mathbb X} _i)\in \{ 1,{\mathfrak{c}}\}$), for each $i\in I$. ${\mathcal T}$ is called actually Vaught's iff for some $\sum _{\mathbb I}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ there are sentences $τ_i\in \mathop{\rm Th}\nolimits ({\mathbb X} _i)$, $i\in I$, providing VC. We prove that: (1) VC is true for ${\mathcal T}$ iff ${\mathcal T}$ is large or its atomic model has a VC decomposition; (2) VC is true for each actually Vaught's FLD$_1$ theory; (3) VC$^\sharp$ is true for ${\mathcal T}$, if there is a VC$^\sharp$-decomposition of a model of ${\mathcal T}$. VC is true for the partial orders from the closure $\langle {\mathcal C} ^{\rm reticle}_0\cup {\mathcal C} ^{\rm ba}\rangle _Σ$, where $\langle {\mathcal C}\rangle _Σ$ denotes the closure of a class ${\mathcal C}$ under finite lexicographic sums. VC$^{\sharp}$ is true for a large class of partial orders of the form $\sum _{\mathbb I}(\dot{\bigcup}_{j<n_i}\prod _{k<m_i^j}{\mathbb X} _i^{j,k})_r$, where ${\mathbb X} _i^{j,k}$-s can be linear orders, or Boolean algebras, or belong to a wide class of trees.

math.LO↗

arXiv2024

Closed sets in a generalization of the Alexandrov cube

We continue work on the topology obtained by the convergence $λ_{ls}$, which started in \cite{KuPaCZ}, and further investigated in \cite{KuPaFil19}. The main goal is to describe the closed sets and closure operator by the family of its minimal elements, with the accent on complete Boolean algebras satisfying countable chain condition.

math.GN↗

arXiv2024

Partial Orders of Bijectively Related or Homeomorphic Topologies

Topologies $τ, σ\in \mathop{\mathrm{Top}}\nolimits _X$ are bijectively related, in notation $τ\sim σ$, if there are continuous bijections $f: (X, τ)\rightarrow (X, σ)$ and $g: (X, σ)\rightarrow (X, τ)$. Defining $[τ]_{\cong}=\{ σ\in \mathop{\mathrm{Top}}\nolimits _X : σ\cong τ\}$ and $[τ]_{\sim }=\{ σ\in \mathop{\mathrm{Top}}\nolimits _X : σ\sim τ\}$ we show that for each infinite 1-homogeneous linear order ${\mathbb L}$ there is a topology $τ\in \mathop{\mathrm{Top}}\nolimits _{|L|}$ such that: (a) $\langle [τ]_{\cong}, \subset \rangle \cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$ (the disjoint union of $2^{|L|}$-many copies of ${\mathbb L}$); so, each maximal chain in $[τ]_{\cong}$ is isomorphic to ${\mathbb L}$; (b) $\langle [τ]_{\sim}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}\widetilde{\mathbb L}$, where $\widetilde{\mathbb L}$ is the Dedekind completion of ${\mathbb L}$; thus, each maximal chain in $[τ]_{\sim}$ is isomorphic to $\widetilde{\mathbb L}$. If, in addition, the linear order ${\mathbb L}$ is Dedekind complete, then the topology $τ$ is weakly reversible, non-reversible and $\langle [τ]_{\sim}, \subset \rangle=\langle [τ]_{\cong}, \subset \rangle\cong \dot{\bigcup}_{2^{|L|}}{\mathbb L}$.

math.GN↗

arXiv2024

Weakly and Strongly Reversible Spaces

A topological space ${\mathcal X}$ is reversible iff each continuous bijection (condensation) $f: {\mathcal X} \rightarrow {\mathcal X}$ is a homeomorphism; weakly reversible iff whenever ${\mathcal Y}$ is a space and there are condensations $f:{\mathcal X} \rightarrow {\mathcal Y}$ and $g:{\mathcal Y} \rightarrow {\mathcal X}$, there is a homeomorphism $h:{\mathcal X} \rightarrow {\mathcal Y}$; strongly reversible iff each bijection $f: {\mathcal X} \rightarrow {\mathcal X}$ is a homeomorphism. We show that the class of weakly reversible non-reversible spaces is disjoint from the class of sequential spaces in which each sequence has at most one limit (containing e.g. metrizable spaces). On the other hand, the class of strongly reversible topologies contains only discrete topologies, antidiscrete topologies and natural generalizations of the cofinite topology.

math.GN↗

arXiv2024

Copies of Monomorphic Structures

The poset of copies of a relational structure ${\mathbb X}$ is the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X})=\{ Y\subset X: {\mathbb Y} \cong {\mathbb X}\}$. Investigating the classification of structures related to isomorphism of the Boolean completions ${\mathbb B}_{\mathbb X} ={\mathop{\rm ro}\nolimits}({\mathop{\rm sq}\nolimits} ({\mathbb P} ({\mathbb X}) ))$ we extend the results concerning linear orders to the class of structures definable in linear orders by first-order $Σ_0$-formulas (monomorphic structures). So, ${\mathbb B}_{\mathbb X} \cong {\mathbb B}_{\mathbb L}$ holds for some linear order ${\mathbb L}$, if ${\mathbb X}$ is definable in a $σ$-scattered (in particular, countable) or additively indecomposable linear order. For example, ${\mathbb B}_{\mathbb X} \cong {\mathop{\rm ro}\nolimits}({\mathbb S} )$, where ${\mathbb S}$ is the Sacks forcing, whenever ${\mathbb X}$ is a non-constant structure chainable by a real order type containing a perfect set.

math.LO↗

arXiv2024

Forcing With Copies of Uncountable Ordinals

For a relational structure ${\mathbb X}$ we investigate the partial order $\langle {\mathbb P} ({\mathbb X}) ,\subset \rangle$, where ${\mathbb P} ({\mathbb X}):=\{ f[X]: f\in \mathop{\rm Emb}\nolimits ({\mathbb X})\}$. Here we consider uncountable ordinals. Since $\mathop{\rm sq}\nolimits {\mathbb P} (α)$ is isomorphic to the direct product $\prod _{i=1}^n (\mathop{\rm sq}\nolimits {\mathbb P} (ω^{δ_i}))^{s_i}$, where $α= ω^{δ_n}s_n+\dots +ω^{δ_1}s_1+ m$ is the Cantor normal form for $α$, the analysis is reduced to the investigation of the posets of the form ${\mathbb P} (ω^{δ})$. It turns out that, in ZFC, either the poset $\mathop{\rm sq}\nolimits {\mathbb P} (α)$ is $σ$-closed and completely embeds $P(ω)/\mathop{\rm Fin}$ and, hence, preserves $ω_1$ and forces $|{\mathfrak c}|=|{\mathfrak h}|$, or, otherwise, completely embeds the algebra $P(λ)/[λ]^{<λ}$, for some regular $ω<λ\leq \mathop{\rm cf}\nolimits (δ)$, and collapses $ω_2$ to $ω$. Regarding the Cantor normal form, the first case appears iff for each $i\leq n$ we have $\mathop{\rm cf}\nolimits (δ_i)\leq ω$, or $δ_i = θ_i + \mathop{\rm cf}\nolimits (δ_i )$, where $\mathop{\mathrm{Ord}}\nolimits \niθ_i \geq \mathop{\rm cf}\nolimits (δ_i ) >\mathop{\rm cf}\nolimits (θ_i )=ω$ and $θ_i =\lim _{n\rightarrow ω}δ_n$, where $\mathop{\rm cf}\nolimits (δ_n)=\mathop{\rm cf}\nolimits (δ_i)$, for all $n\in ω$.

math.LO↗

arXiv2024

Reduced Products of Collapsing Algebras

$\mathop{\rm rp}\nolimits ({\mathbb B})$ denotes the reduced power ${\mathbb B}^ω/Φ$ of a Boolean algebra ${\mathbb B}$, where $Φ$ is the Fréchet filter $Φ$ on $ω$. We investigate iterated reduced powers ($\mathop{\rm rp}\nolimits ^0 ({\mathbb B})={\mathbb B}$ and $\mathop{\rm rp}\nolimits ^{n+1} ({\mathbb B} )=\mathop{\rm rp}\nolimits (\mathop{\rm rp}\nolimits ^n ({\mathbb B}))$) of collapsing algebras and our main intention is to classify the algebras $\mathop{\rm rp}\nolimits ^n (\mathop{\rm Col}\nolimits (λ,κ))$, $n\in {\mathbb N}$, up to isomorphism of their Boolean completions. In particular, assuming that SCH and ${\mathfrak h} =ω_1$ hold, we show that for any cardinals $λ\geq ω$ and $κ\geq 2$ such that $κλ>ω$ and $\mathop{\rm cf}\nolimits (λ)\leq {\mathfrak c}$ we have $\mathop{\rm ro} (\mathop{\rm rp}\nolimits ^n(\mathop{\rm Col}\nolimits (λ,κ)))\cong \mathop{\rm Col}\nolimits (ω_1, (κ^{<λ})^ω)$, for each $n\in {\mathbb N}$. If ${\mathfrak b} ={\mathfrak d}$ and $0^\sharp$ does not exist, then the same holds whenever $\mathop{\rm cf}\nolimits (λ)= ω$.

math.LO↗

arXiv2023

Reversible and Irreversible Trees

A tree ${\mathbb T} =\langle T\leq \rangle$ is reversible iff there is no order $\preccurlyeq \;\varsubsetneq \;\leq $ such that ${\mathbb T} \cong \langle T ,\preccurlyeq\rangle$. Using a characterization of reversibility via back and forth systems we detect a wide class of non-reversible trees: ``bad trees" (having all branches of height ${\mathrm{ht}} ({\mathbb T})=|T|=|L_0|$, where $|T|$ is a regular cardinal). Consequently, a countable tree of height $ω$ and without maximal elements is reversible iff all its nodes are finite. We show that a tree ${\mathbb T}$ is non-reversible iff it contains a ``critical node" or an ``archetypical subtree" (parts of ${\mathbb T}$ with some combinatorial properties). In particular, a tree with finite nodes ${\mathbb T}$ is reversible iff it does not contain archetypical subtrees. Using that characterization we prove that if for each ordinal $α\in [ω,{\mathrm{ht}} ({\mathbb T}))$ all nodes of height $α$ are of the same size, or the sequence $\langle \langle |N|,|N\uparrow|\rangle : {\mathcal{N}} ({\mathbb T}) \ni N\subset L_α\rangle $ is finite-to-one, then ${\mathbb T}$ is reversible. Consequently, regular $n$-ary trees are reversible, reversible Aronszajn trees exist and, if there are Suslin or Kurepa trees, there are reversible ones. Also we show that for cardinals $λ>1$ and $μ>0$ and ordinal $α>0$ we have: the tree $\bigcup _μ{}^{<α}λ$ is reversible iff $\min\{α,λμ\} <ω$.

math.LO↗

arXiv2023

Posets of Copies of Countable Ultrahomogeneous Tournaments

The poset of copies of a relational structure ${\mathbb X}$ is the partial order ${\mathbb P} ({\mathbb X} ) := \langle \{ Y \subset X: {\mathbb Y} \cong {\mathbb X}\}, \subset \rangle$ and each similarity of such posets (e.g. isomorphism, forcing equivalence) determines a classification of structures. We consider the countable ultrahomogeneous tournaments: ${\mathbb Q} $ (the rational line), ${\mathbb S} (2)$ (the circular tournament), and ${\mathbb T} ^\infty$ (the random tournament); as well as the ultrahomogeneous digraphs ${\mathbb S} (3)$, ${\mathbb Q} [{\mathbb I}_n]$, ${\mathbb S} (2)[{\mathbb I}_n]$ and ${\mathbb T} ^\infty [{\mathbb I}_n]$ from Cherlin's list. If ${\mathbb G} _{\mathrm{Rado}}$ (resp. ${\mathbb Q} _n$) denotes the countable homogeneous universal graph (resp. $n$-labeled linear order), it turns out that ${\mathbb P} ({\mathbb T} ^\infty)\cong {\mathbb P} ({\mathbb G}_{\mathrm{Rado}})$ and that ${\mathbb P} ({\mathbb Q} _n)$ densely embeds in ${\mathbb P} ({\mathbb S} (n))$, for $n\in\{ 2,3\}$. Consequently, ${\mathbb B} _{\mathbb X} \cong {\mathrm{ro}}\, ({\mathbb S} \ast π)$, where ${\mathbb S}$ is the Sacks forcing and $1_{\mathbb S} \Vdash "π$ is a separative, atomless and $σ$-closed forcing", whenever ${\mathbb X}$ is a countable structure equimorphic with ${\mathbb Q}$, ${\mathbb Q} _n$, ${\mathbb S} (2)$, ${\mathbb S} (3)$, ${\mathbb Q} [{\mathbb I}_n]$ or ${\mathbb S} (2)[{\mathbb I}_n]$. Also, ${\mathbb B} _{\mathbb X} \cong {\mathrm{ro}}\, ({\mathbb S} \ast π)$, where $1_{\mathbb S} \Vdash "π$ is an $ω$-distributive forcing", whenever ${\mathbb X}$ is a countable graph embedding ${\mathbb G} _{\mathrm{Rado}}$, or a countable tournament embedding ${\mathbb T} ^\infty$, or ${\mathbb X} ={\mathbb T} ^\infty [{\mathbb I}_n]$.

math.LO↗

arXiv2023

Sharp Vaught's Conjecture for Some Classes of Partial Orders

Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, $I({\mathcal T},ω)\in \{ 0,1,{\mathfrak{c}}\}$, holds for each complete theory of linear order ${\mathcal T}$. We show that the same is true for each complete theory of partial order having a model in the the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension of the class of rooted trees admitting a finite monomorphic decomposition, obtained in the same way. The sharp version of Vaught's conjecture also holds for the theories of trees which are infinite disjoint unions of linear orders.

math.LO↗

arXiv2023

Back and Forth Systems Witnessing Irreversibility

If $L$ is a relational language, then an $L$-structure ${\mathbb X}=\langle X,\bar ρ\rangle$ is reversible iff there is no interpretation $\bar σ\varsubsetneq \bar ρ$ such that the structures $\langle X,\bar σ\rangle$ and $\langle X,\bar ρ\rangle$ are isomorphic. We show that ${\mathbb X}$ is not reversible iff there is a back and forth system $Π$ of partial self-condensations of ${\mathbb X}$ containing one which is not a partial isomorphism and having certain closure properties. Using that characterization we detect several classes of non-reversible partial orders containing, for example, homogeneous-universal posets (in particular, the random poset), the divisibility lattice, $\langle {\mathbb N} ,\,\mid\,\rangle$, the ideals $[κ]^{<λ}$, the meager ideal in the algebra Borel$(ω^ω)$, and the direct powers of rationals, ${\mathbb Q} ^κ$, and integers, ${\mathbb Z} ^κ$. Some of the results are obtained under additional set-theoretic assumptions.

math.LO↗

arXiv2022

Connected Reduced Products

If $ρ$ is a binary relation on a set $X$, the structure ${\mathbb X}=\langle X,ρ\rangle$ is connected iff the minimal equivalence relation containing $ρ$ is the full relation on $X$. We show that, for a set $I$ the following conditions are equivalent (a) $|I|$ is less than the first measurable cardinal, (b) For each filter $Φ\subset P(I)$ and each family $\{ {\mathbb X}_i :i\in I\}$ of binary structures, the reduced product $\prod_Φ{\mathbb X}_i$ is connected, iff there are a finite set $K\subset I$ and $n\in ω$ such that ${\mathbb X}_i$ is connected, for each $i\in K$, and $\{ i\in I: {\mathbb X}_i \mbox{ is of diameter }\leq n\}\cup K\in Φ$, (c)The ultraproduct $\prod_{\mathcal U}{\mathbb G}_ω$ is a disconnected graph for each non-principal ultrafilter ${\mathcal U}\subset P(I)$, where ${\mathbb G}_ω$ is the linear graph on $ω$. Moreover, the implication "$\Leftarrow$" in (b) holds in ZFC.

math.LO↗

arXiv2019

Vaught's Conjecture for Almost Chainable Theories

A structure ${\mathbb Y}$ of a relational language $L$ is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $<$ on the set $Y\setminus F$ such that for each partial automorphism $φ$ (i.e., local automorphism, in Fra\"ıssé's terminology) of the linear order $\langle Y\setminus F, < \rangle$ the mapping ${\mathrm{id}} _F \cup φ$ is a partial automorphism of ${\mathbb Y}$. By a theorem of Fra\"ıssé, if $|L|<ω$, then ${\mathbb Y}$ is almost chainable iff the profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer $m$ such that ${\mathbb Y}$ has $\leq m$ non-isomorphic substructures of size $n$, for each positive integer $n$. A complete first order $L$-theory ${\mathcal T}$ having infinite models is called almost chainable iff all models of ${\mathcal T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of ${\mathcal T}$. In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

math.LO↗

arXiv2019

Antichains of Copies of Ultrahomogeneous Structures

We investigate possible cardinalities of maximal antichains in the poset of copies $\langle \mathbb P(\mathbb X),\subset \rangle$ of a countable ultrahomogeneous relational structure $\mathbb X$. It turns out that if the age of $\mathbb X$ has the strong amalgamation property, then, defining a copy of $\mathbb X$ to be large iff it has infinite intersection with each orbit of $\mathbb X$, the structure $\mathbb X$ can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum in the poset $\mathbb P (\mathbb X)$. Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the random (universal ultrahomogeneous) poset.

math.LO↗

arXiv2018

Vaught's Conjecture for Monomorphic Theories

A complete first order theory of a relational signature is called monomorphic iff all its models are monomorphic (i.e. have all the $n$-element substructures isomorphic, for each positive integer $n$). We show that a complete theory ${\mathcal T}$ having infinite models is monomorphic iff it has a countable monomorphic model and confirm the Vaught conjecture for monomorphic theories. More precisely, we prove that if ${\mathcal T}$ is a complete monomorphic theory having infinite models, then the number of its non-isomorphic countable models, $I({\mathcal T},ω)$, is either equal to $1$ or to ${\mathfrak c}$. In addition, $I({\mathcal T},ω)= 1$ iff some countable model of ${\mathcal T}$ is simply definable by an $ω$-categorical linear order on its domain.

math.LO↗

arXiv2018

The Left, the Right and the Sequential Topology on Boolean Algebras

For the algebraic convergence $λ_{\mathrm{s}}$, which generates the well known sequential topology $τ_s$ on a complete Boolean algebra ${\mathbb B}$, we have $λ_{\mathrm{s}}=λ_{\mathrm{ls}}\cap λ_{\mathrm{li}}$, where the convergences $λ_{\mathrm{ls}}$ and $λ_{\mathrm{li}}$ are defined by $λ_{\mathrm{ls}}(x)=\{ \limsup x\}\!\uparrow$ and $λ_{\mathrm{li}}(x)=\{ \liminf x\}\!\downarrow$ (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology $\mathcal{O}_{\mathrm{lsi}}$ extending the (unique) sequential topologies $\mathcal{O}_{λ_{\mathrm{ls}}}$ (left) and $\mathcal{O}_{λ_{\mathrm{li}}}$ (right) generated by the convergences $λ_{\mathrm{ls}}$ and $λ_{\mathrm{li}}$ and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences. In addition, we observe some special classes of algebras and, in particular, show that in $(ω,2)$-distributive algebras we have $\lim_{{\mathcal O}_{\mathrm{lsi}}}=\lim_{τ_{\mathrm{s}} }=λ_{\mathrm{s}}$, while the equality $\mathcal{O}_{\mathrm{lsi}}=τ_s$ holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity.

math.GN↗