Isomorphic Copy Research Articles

Embedding the free topological group F(Xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X^n)$$\\end{document} into F(X)

In 1976, Nickolas showed that for each natural n, the free topological group F(Xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X^n)$$\\end{document} is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a kω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\\omega }$$\\end{document}-space. We complement the Nickolas’ embedding theorem by showing that it remains true for every topological space X such that all finite powers of X are pseudocompact. For example, all pseudocompact k-spaces enjoy this property. Also, we extend the embedding theorem to the class of NCω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$NC_\\omega $$\\end{document}-spaces that includes, in particular, the kω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_\\omega $$\\end{document}-spaces and the well-ordered spaces of ordinals [0,α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0, \\alpha )$$\\end{document}, for every ordinal α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Our results are quite sharp because we present a first example of a Tychonoff space Z such that F(Z) does not contain an isomorphic copy of the group F(Z2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(Z^2)$$\\end{document}. In addition, our space Z is countably compact, separable, and its square Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z^2$$\\end{document} is not pseudocompact.

Minimal monoids generating varieties with complex subvariety lattices

AbstractA variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. We show that the 6-element Brandt monoid generates a finitely universal variety of monoids and, by the previous results, it is the smallest generator for a monoid variety with this property. It is also deduced that the join of two Cross varieties of monoids can be finitely universal. In particular, we exhibit a finitely universal variety of monoids with uncountably many subvarieties which is the join of two Cross varieties of monoids whose lattices of subvarieties are the 6-element and the 7-element chains, respectively.

A quasidouble of the affine plane of order 4 and the solution of a problem on additive designs

A 2-(v,k,λ) block design (P,B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G,+) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G, the embedding of P in G is also such that, conversely, any zero-sum k-subset of P is a block in B, then (P,B) is said to be strongly additive.In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2-(16,4,2) design (F4×F4,B2), which decomposes into two disjoint isomorphic copies of the affine plane of order four.An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG(4,2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem.

Finite Idempotent Set-Theoretic Solutions of the Yang–Baxter Equation

Abstract It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang–Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $\varphi _{x}$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated Yang–Baxter monoid $M(X,r)$ is cancellative and all the maps $\varphi _{x}$ are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_{u}$ indexed by the diagonal, each $S_{u}$ has a group of quotients $G_{u}$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.

COFINAL TYPES BELOW

Abstract It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph _n$ is at least $c_{n+2}$ , the $(n+2)$ -Catalan number. Moreover, the class $\mathcal D_{\aleph _n}$ of directed sets of cardinality $\leq \aleph _n$ contains an isomorphic copy of the poset of Dyck $(n+2)$ -paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

The Hole Argument and Beyond: Part II: Treating Non-isomorphic Spacetimes

In this two-part paper we review, and then develop, the assessment of the hole argument for general relativity.The review (in Part I) discussed how to compare points in isomorphic spacetimes, i.e. models of the theory. This second Part proposes a framework for making comparisons of non-isomorphic spacetimes. It combines two ideas we discussed in Part I—the philosophical idea of counterparts, and the idea of threading points between spacetimes other than by isomorphism—with the mathematics of fibre bundles.We first recall the ideas from Part I (Section 1). Then in Section 2 and an Appendix, we define a fibre bundle whose fibres are isomorphic copies of a given spacetime or model, and discuss connections on this fibre bundle. This material proceeds on analogy with field-space formulations of gauge theories. Finally, in Section 3, we show how this fibre bundle gives natural expressions of the philosophical ideas of counterparts, and of threading.

Topological group actions by group automorphisms and Banach representations

Abstract We study Banach representability for actions of topological groups on groups by automorphisms (in particular, an action of a group on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 ⁡ ( ℝ ) {\operatorname{SL}_{2}(\mathbb{R})} . The conjugation action of SL n ⁡ ( ℝ ) {\operatorname{SL}_{n}(\mathbb{R})} is not Asplund representable for every n ≥ 4 {n\geq 4} . The linear action of GL n ⁡ ( ℝ ) {\operatorname{GL}_{n}(\mathbb{R})} on ℝ n {{\mathbb{R}}^{n}} , for every n ≥ 2 {n\geq 2} , is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 {l_{1}} ). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). As a byproduct, we obtain some counterexamples about Banach representations of homogeneous G-actions G / H {G/H} .

Isomorphic Copies of ell ^infty in the Weighted Hardy Spaces on the Unit Disc

It is still unclear whether the density of analytic polynomials in an H-admissible space is sufficient to the minimality of the space? This question has a purely foundational background, relating fundamental concepts from the theory of H^p spaces. We hypothesize that there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space. We solve this problem by finding suitable counterexamples of Hardy spaces built upon some weighted Lebesgue spaces. In particular, we provide a direct construction of weights from Szegő class, which guarantees the existence of isomorphic copies of the space of bounded sequences in weighted Hardy spaces on the unit disc.

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

AbstractLet be a class of graphs with a membership test, , and let be the class of graphs in of path‐width at most . We present an interactive framework that finds an unavoidable set for , which is a set of graphs such that any graph in contains an isomorphic copy of a graph in . At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for . While obstruction sets are well‐studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever is locally checkable of bounded maximum degree and is a finite set of connected graphs. For example, ‐regular graphs, ‐colourable graphs, and ‐free graphs are locally checkable classes. We put special emphasis on the case that is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high‐degree‐first path‐decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path‐width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path‐width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path‐width 3 and girth 4.

Disjoint isomorphic balanced clique subdivisions

A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least Ck2 has a subdivision of Kk, the complete graph on k vertices. We study two directions extending this result.•Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a Kk-subdivision.•Thomassen conjectured that for each k∈N there is some d=d(k) such that every graph with average degree at least d contains a balanced subdivision of Kk. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on d(k) remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced Kk-subdivision. This gives the right order of magnitude of the optimal d(k) needed in Thomassen's conjecture. Since a balanced Kmk-subdivision trivially contains m vertex-disjoint isomorphic Kk-subdivisions, this also confirms Verstraëte's conjecture in a strong sense.

Isomorphisms between random graphs

Consider two independent Erdős–Rényi G(N,1/2) graphs. We show that with probability tending to 1 as N→∞, the largest induced isomorphic subgraph has size either ⌊xN−εN⌋ or ⌊xN+εN⌋, where xN=4log2⁡N−2log2⁡log2⁡N−2log2⁡(4/e)+1 and εN=(4log2⁡N)−1/2. Using similar techniques, we also show that if Γ1 and Γ2 are independent G(n,1/2) and G(N,1/2) random graphs, then Γ2 contains an isomorphic copy of Γ1 as an induced subgraph with high probability if n≤⌊yN−εN⌋ and does not contain an isomorphic copy of Γ1 as an induced subgraph with high probability if n>⌊yN+εN⌋, where yN=2log2⁡N+1 and εN is as above.

Transfinite almost square Banach spaces

Is is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose of relating them to the

(1+)-complemented, (1+)-isomorphic copies of L_{1} in dual Banach spaces

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler–Stegall characterisation of dual spaces containing complemented copies of L_{1}. As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space X, the following statements are equivalent:X contains almost isometric contains almost isometric copies of (bigoplus _{n=1}^{infty } ell _{infty }^{n})_{ell _1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of L_{1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of C[0,1]^{*}. Moreover, if X is separable, one may add the following assertion:for all varepsilon >0, there exists a (1+varepsilon )-quotient map T:Xrightarrow C(Delta ) so that T^{*}[C(Delta )^{*}] is (1+varepsilon )-complemented in X^{*}, where Delta is the Cantor set

Zero-Sum Copies of Spanning Forests in Zero-Sum Complete Graphs

For a complete graph \(K_n\) of order n, an edge-labeling \(c:E(K_n)\rightarrow \{ -1,1\}\) satisfying \(c(E(K_n))=0\), and a spanning forest F of \(K_n\), we consider the problem to minimize \(|c(E(F'))|\) over all isomorphic copies \(F'\) of F in \(K_n\). In particular, we ask under which additional conditions there is a zero-sum copy, that is, a copy \(F'\) of F with \(c(E(F'))=0\). We show that there is always a copy \(F'\) of F with \(|c(E(F'))|\le \Delta (F)+1\), where \(\Delta (F)\) is the maximum degree of F. We conjecture that this bound can be improved to \(|c(E(F'))|\le (\Delta (F)-1)/2\) and verify this for F being the star \(K_{1,n-1}\). Under some simple necessary divisibility conditions, we show the existence of a zero-sum \(P_3\)-factor, and, for sufficiently large n, also of a zero-sum \(P_4\)-factor.

Orthogonal Labeling for Some Different Infinite Graph Classes

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s. Graph labeling is one of the fascinating areas of graph theory with wide ranging applications. A collection $\mathcal{A}$ of isomorphic copies of a given subgraph $G$ of $T$ is said to be an orthogonal double cover (ODC) of the graph $T$ by $G$, if every edge of $T$ belongs to exactly two members of $\mathcal{A}$ and any two different elements from $\mathcal{A}$ share at most one edge. The existence of an orthogonal labeling of a graph $G$ with respect to a certain group implies the existence of the cyclic orthogonal double cover of the circulant graphs on that group. In this paper, we prove the existence of orthogonal labeling for some different infinite graph classes and hence, the existence of the cyclic orthogonal double cover of some different infinite circulant graphs.

Tree embeddings and tree‐star Ramsey numbers

AbstractWe say that a graph can be embedded into a graph if contains an isomorphic copy of as a subgraph. Guo and Volkmann conjectured that if is a connected graph with at least vertices and minimum degree at least , then any tree with vertices and maximum degree at most can be embedded into . In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with vertices and maximum degree at most can be embedded into a connected graph G with at least vertices and minimum degree at least . Our result implies that the conjecture of Guo and Volkmann is true with one exception. We also give an application to the Ramsey number of a tree versus a star.

Weak limited sets and operators on Banach lattices

In this paper, we prove that an operator T : E → F , between two Banach lattices, maps order intervals onto weak limited sets if and only if the modulus |ST| exists and is Dunford-Pettis for every Dunford-Pettis operator S : F → c0. Next, we establish that a Banach lattice E does not contain any isomorphic copy of ℓ 1 if and only if the order intervals of E are weak limited and the norm of E′ is order continuous. We also investigate the domination problem of the class of weak limited operators.

On color isomorphic subdivisions

Given a graph H and an integer k⩾2, let fk(n,H) be the smallest number of colors C such that there exists a proper edge-coloring of the complete graph Kn with C colors containing no k vertex-disjoint color isomorphic copies of H. In this paper, we prove that f2(n,Ht)=Ω(n1+12t−3) where Ht is the 1-subdivision of the complete graph Kt. This answers a question of Conlon and Tyomkyn (2021) [4].

Centralisers of Formal Maps

We consider the formal maps in any finite dimension d with coefficients in an integral domain K with identity. Those invertible under formal composition form a group [inline-graphic 1i]. We consider the centraliser Cg of an element g ∈ [inline-graphic 1i] of infinite order, tangent to the identity of [inline-graphic 1i]. If g has infinite order and K is a field of characteristic zero, we show that Cg contains an isomorphic copy of the additive group (K, +). If g has infinite order and K has positive characteristic, we show that Cg contains an uncountable abelian subgroup. The proofs are quite different in finite characteristic and in characteristic zero, but are connected by so-called sum functions.

Universal groups of cellular automata

We prove that the group of reversible cellular automata (RCA), on any alphabet $A$, contains a subgroup generated by three involutions which contains an isomorphic copy of every finitely generated group of RCA on any alphabet $B$. This result follows from