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Register a new userStrongly Minimal Steiner Systems II: Coordinatization and Quasigroups
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John T. Baldwin
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We note that a strongly minimal Steiner $k$-Steiner system $(M,R)$ from (Baldwin-Paolini 2020) can be `coordinatized in the sense of (Gantner-Werner 1975) by a quasigroup if $k$ is a prime-power. But for the basic construction this coordinatization is never definable in $(M,R)$. Nevertheless, by refining the construction, if $k$ is a prime power there is a $(2,k)$-variety of quasigroups which is strongly minimal and definably coordinatizes a Steiner $k$-system.
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Let $M$ be strongly minimal and constructed by a `Hrushovski construction. If the Hrushovski algebraization function $mu$ is in a certain class ${mathcal T}$ ($mu$ triples) we show that for independent $I$ with $|I| >1$, ${rm dcl}^*(I)= emptyset$ (* means not in ${rm dcl}$ of a proper subset). This implies the only definable truly $n$-ary function $f$ ($f$ `depends on each argument), occur when $n=1$. We prove, indicating the dependence on $mu$, for Hrushovskis original construction and including analogous results for the strongly minimal $k$-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, ${rm sdcl}^*(I) =emptyset$, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k = p^n$. The proofs depend on our introduction for appropriate $G subseteq {rm aut}(M)$ the notion of a $G$-normal substructure ${mathcal A}$ of $M$ and of a $G$-decomposition of ${mathcal A}$. These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit.
The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.
Among other results, we prove the following theorem about Steiner minimal trees in $d$-dimensional Euclidean space: if two finite sets in $mathbb{R}^d$ have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.
Let $X$ be a $v$-set, $B$ a set of 3-subsets (triples) of $X$, and $B^+cupB^-$ a partition of $B$ with $|B^-|=s$. The pair $(X,B)$ is called a simple signed Steiner triple system, denoted by ST$(v,s)$, if the number of occurrences of every 2-subset of $X$ in triples $BinB^+$ is one more than the number of occurrences in triples $BinB^-$. In this paper we prove that $st(v,s)$ exists if and only if $vequiv1,3pmod6$, $v e7$, and $sin{0,1,...,s_v-6,s_v-4,s_v}$, where $s_v=v(v-1)(v-3)/12$ and for $v=7$, $sin{0,2,3,5,6,8,14}$.
A standard tool for classifying the complexity of equivalence relations on $omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $Gamma$ be one of the following classes: $Sigma^0_{alpha}$, $Pi^0_{alpha}$, $Sigma^1_n$, or $Pi^1_n$, where $alpha geq 2$ is a computable ordinal and $n$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $Gamma$.
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