In quantum mechanics, each observable is assigned a collection of projections. Two observables are compatible (can be measured simultaneously) if and only if any two projections that are assigned to them commute. This led to the study of noncommuting sets of idempotents which further led to the skew lattice theory that was founded and explored by Jonathan Leech. In the present paper we shall se...

We study polynomial growth of algebraic series. It is proved that if an Nalgebraic series r with arbitrarily many commuting variables has polynomial growth, then r is in fact N-rational. Also, polynomial growth is decidable. Similar results are proved forN-algebraic series with noncommuting variables having bounded supports. It is also shown that polynomial growth is not decidable for N-algebra...

Consider the algebra Q〈〈x1, x2, . . .〉〉 of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x1, x2, . . .) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In p...

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variab...

The pillars of quantum theory include entanglement and operators' failure to commute. Page curve quantifies the bipartite a many-body system in random pure state. This is known decrease if one constrains extensive observables that commute with each other (Abelian ``charges''). Non-Abelian charges, which fail other, are current interest thermodynamics. For example, noncommuting charges were show...