The lattice of monomial clones on finite fields

We investigate the lattice of clones that are generated by a set of functions that are induced on a finite field $${\mathbb {F}}$$ F by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains.We give a connection between the lattice of these clones and semi-affine algebras. Furthermore, we show that the sublattice of idempotent clones of this lattice is finite and every idempotent monomial clone is principal.

Correlation Functions of Quantum Toroidal $\mathfrak{gl}_1$ Algebra

In this paper, we study the correlation functions of the quantum toroidal $\mathfrak{gl}_1$ algebra. The first key properties we establish are similar to those of the correlation functions of quantum affine algebras $U_q\mathfrak{n}_+$ as established by Enriquez in (Eneiquez, 2000), while the proof of the remaining key ``vanishing property" relies on a certain ``Master Equality'' of formal series, which constitutes the main technical result of this paper.