arXiv · 0706.1190
Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.
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Martin Goldstern, Saharon Shelah. 2007-06-08. All creatures great and small. https://arxiv.org/abs/0706.1190
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Quasi-projective dimension of modules over associative rings is generalized in this paper to the one of complexes of modules. Basic properties of this dimension are established, including a comparison result with projective dimension and a derived Auslander-Buchsbaum formula for complexes of finite quasi-projective dimension. Several sufficient conditions are provided for a commutative noetherian local ring to be a complete intersection under the assumption that each finitely generated module has finite quasi-projective dimension. This provides some positive answers to an open question on quasi-projective dimension proposed by Gheibi-Jorgensen-Takahashi. Moreover, the behavior of quasi-projective dimension under taking the quotient of a commutative ring modulo a regular sequence is investigated, and some partial results toward the change-of-rings question on quasi-projective dimension are given.
Let $ϕ$ be a quaternion of modulus $1$. In this article, we study some topics related to $ϕ$-conjugation for quaternionic matrices, including $ϕ$-Hermitian matrices, $ϕ$-conjugate normal matrices, unitary $ϕ$-congruence and $ϕ$-HSH decomposition (decomposition of a $ϕ$-Hermitian matrix and a skew $ϕ$-Hermitian matrix). In particular, we generalize the Autonne-Takagi factorization of quaternion $ϕ$-Hermitian matrices for all unit quaternion $ϕ$. This gives an affirmative answer to a problem proposed by R. Horn and F. Zhang in the paper ``A generalization of the complex Autonne-Takagi factorization to quaternion matrices, Linear Multilinear A. 60: 1239--1244, 2012''.
We develop an operadic theory of anti-splitting and anti-replication for binary operads. For a binary operad $\opd$, we define its anti-bisuccessor $\ABSu(\opd)$ and anti-trisuccessor $\ATSu(\opd)$, with the property that an operation of $\opd$ splits into a negative sum of two or three operations, yielding canonical operad morphisms $\opd \to\ABSu(\opd)$ and $\opd\to\ATSu(\opd)$. For binary quadratic operads, the process gives Manin black products, \[ \ABSu(\opd) \cong \antiprelie \bullet \opd, \qquad \ATSu(\opd)\cong \antipostlie\bullet\opd. \] Dually, we introduce the anti-duplicator \(\ADup(\mathcal P)\) and anti-triplicator \(\ATrip(\mathcal P)\) and prove \[ \ADup(\opd^!)\cong \ABSu(\opd)^!, \qquad \ATrip(\opd^!)\cong \ATSu(\opd)^!. \] Consequently, the anti-duplicator gives a new way to calculate Manin white products. %, in particular, $\ADup(\opd)\cong \antiperm\circ\opd$. The framework recovers known anti-splittings, including anti-dendriform, anti-pre-Lie, and anti-pre-Poisson structures, in a uniform operadic form and produces further examples and dual pairs.