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For a numerical semigroup S $subseteq$ N with embedding dimension e, conductor c and left part L = S $cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $ge$ 0 always holds, a question known since as Wilfs conjecture. Using a closely related lower bound W 0 (S) $le$ W (S), we show that if |L| $le$ 12 then W 0 (S) $ge$ 0, thereby settling Wilfs conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = --1. Wilfs conjecture remains open for |L| $ge$ 13.
In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that this class
Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is cyclically fully commutative (CFC) if every cyclic shift of any
We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphi
Pairwise non-isomorphic semigroups obtained from the finite inverse symmetric semigroup $mathcal{IS}_n ,$ finite symmetric semigroup $mathcal{T}_n$ and bicyclic semigroup by the deformed multiplication proposed by Ljapin are classified.
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In additi