Related papers: Small clones and the projection property

Let X be a set and Bin(X) the set of all binary operations on X. We say that a subset of Bin(X) is a distributive set of operations if all pairs of elements are right distributive. J.Przytycki posed the question of which groups can be…

Quantum cloning is a fundamental protocol of quantum information theory. Perfect universal quantum cloning is prohibited by the laws of quantum mechanics, only imperfect copies being reachable. Symmetric quantum cloning is concerned with…

Due to the Baker-Pixley theorem we know that every clone over a finite domain $A$ containing a near-unanimity operation $g$ is finitely generated. Therefore there exists an integer $k$ such that the clone is generated by its $k$-ary part.…

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show…

Let $G$ be a Lie group $G$ with representation $\rho$ on a real simple $G$-module $\mathbb{V}$. We will call the orbits of the induced action of $\rho$ on the projectivization $P\mathbb{V}$ the projective orbits, and projective orbits of…

We obtain recurrences for smallest parts functions which resemble Euler's recurrence for the ordinary partition function. The proofs involve the holomorphic projection of non-holomorphic modular forms of weight 2.

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that…

We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We…

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions for certain pairs $(C_1,C_2)$ of clones of Boolean functions. Namely, when $C_1$ is a subclone (a proper subclone, resp.)…

We study functional clones, which are sets of non-negative pseudo-Boolean functions (functions $\{0,1\}^k\to\mathbb{R}_{\geq 0}$) closed under (essentially) multiplication, summation and limits. Functional clones naturally form a lattice…

We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new…

Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…

Let $(G,X)$ be a Shimura pair of Hodge type such that $G$ is the Mumford--Tate group of some elements of $X$. We assume that for each simple factor $G_0$ of $G^{\ad}$ there exists a simple factor of $G_{0\dbR}$ which is compact. Let $N\Ge…

Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\mathbf A$ and $\mathbf B$ there are only finitely…

Fix a manifold M, and let V be an infinite dimensional Lie algebra of vector fields on M. Assume that V contains a finite dimensional semisimple maximal subalgebra A, the projective or conformal subalgebra. A projective or conformal…

We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$. There are a total of 70 UMTCs of rank $\leq 4$ (Note that some authors would have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs…

Every clone of functions comes naturally equipped with a topology---the topology of pointwise convergence. A clone $\mathfrak{C}$ is said to have automatic homeomorphicity with respect to a class $\mathcal{C}$ of clones, if every…

In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\M$ and $\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\M$ is…

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper…

For a countable, weakly minimal theory, we show that the Schroeder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to both a condition on orbits of rank 1 types and the property that the theory…