Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigro...

There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F ). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois g...

This paper is concerned with the conjectural correspondence between Galois representations and modular forms. Although such relation has been fully established in the case the Galois representation is realizable on the `-adic Tate module of an elliptic curve over Q (cf. [24],[20],[5]), very little is known in general. As a first step towards the understanding of more complicated cases, we consi...

We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl(A) and closed subgroups of the Galois group Aut(acl(A)/A). We also characterize when the topology of the Ga...

There is a serious discrepancy among literature on the Picard-Vessiot theory in positive characteristics (for iterative differential fields). We consider descriptions of Galois correspondence in four approaches to this subject: Okugawa’s result [7], Takeuchi’s Hopf algebraic approach [11] (and [3]), the result of Matzat and van der Put [6], and the model theoretic approach by Pillay [8]. In the...

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p : (Q̃, q̃) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q) := Aut(p) = {g ∈ Adj(Q)′ | qg = q}, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected c...