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Spectral Real Semigroups

M. Dickmann, A. Petrovich (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the...

Squarefree monomial ideals with maximal depth

Ahad Rahimi (2020)

Czechoslovak Mathematical Journal

Let ( R , 𝔪 ) be a Noetherian local ring and M a finitely generated R -module. We say M has maximal depth if there is an associated prime 𝔭 of M such that depth M = dim R / 𝔭 . In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.

Stable short exact sequences and the maximal exact structure of an additive category

Wolfgang Rump (2015)

Fundamenta Mathematicae

It was recently proved that every additive category has a unique maximal exact structure, while it remained open whether the distinguished short exact sequences of this canonical exact structure coincide with the stable short exact sequences. The question is answered by a counterexample which shows that none of the steps to construct the maximal exact structure can be dropped.

Standard monomials for q-uniform families and a conjecture of Babai and Frankl

Gábor Hegedűs, Lajos Rónyai (2003)

Open Mathematics

Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family ℱ k , q = K ⊆ n : K ≡ k ( m o d q ) We study certain inclusion matrices attached to F(k,q) over the field 𝔽 p . We show that if l≤q−1 and 2l≤n then r a n k 𝔽 p I ( ℱ ( k , q ) , n ⩽ ℓ ) ⩽ n ℓ This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.

Stanley decompositions and polarization

Sarfraz Ahmad (2011)

Czechoslovak Mathematical Journal

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal I is a CM Stanley ideal, then I p is a Stanley ideal as well, where I p is the polarization of I .

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