Algebraic structures in which the property of commutativity is substituted by mediality are introduced. We consider (associative) graded algebras and instead almost (generalized or $\varepsilon$-commutativity) we introduce ("commutativity-to-mediality" ansatz). Higher twisted products "deforming" brackets (being medial analog Lie brackets) defined. Toyoda's theorem connects (universal) with abe...

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...

The necessary and sufficient conditions under which a Menger algebra can be isomorphically represented by medial n-ary operations are proposed. Since hypercompositional regarded as generalization of algebra, for this reason, the situation hyperoperations is further examined representation theorem algebras such concepts proved.

Factoring a Synchronous Context-Free Grammar into an equivalent grammar with a smaller number of nonterminals in each rule enables more efficient strategies for synchronous parsing. We present an algorithm for factoring an n-ary SCFG into a k-ary grammar in time O(kn). We also show how to efficiently compute the exact number of k-ary parsable permutations of length n, and discuss asymptotic beh...

Fundamentally binary theories are nonsignaling theories in which measurements of many outcomes are constructed by selecting from binary measurements. They constitute a sensible alternative to quantum theory and have never been directly falsified by any experiment. Here we solve two open problems related to them raised in Phys. Rev. Lett. 117, 150401 (2016). We first show that fundamentally bina...

It is well known that the k-ary n-cube has been one of the most efficient interconnection networks for distributed-memory parallel systems. A k-ary n-cube is bipartite if and only if k is even. Let (X,Y ) be a bipartition of a k-ary 2-cube (even integer k ≥ 4). In this paper, we prove that for any two healthy vertices u ∈ X, v ∈ Y , there exists a hamiltonian path from u to v in the faulty k-ar...

Modern question answering and summarizing systems have motivated the need for complex n-ary relation extraction systems where the number of related entities (n) can be more than two. Shortest path dependency kernels have been proven to be effective in extracting binary relations. In this work, we propose a method that employs shortest path dependency based rules to extract complex n-ary relatio...