Quandles, quasoids and projections

We study connections between two algebraic constructions — quandles and quasoids. Both of them are motivated by colorings of knot diagrams. There are constructions of cocyclic knot invariants as for quandles and for quasoids. In this paper we, at first, introduce the notion of projection from quasoid to quandle and, at second, construct mixed cocyclic invariants, which use quandles, quasoids and projections in their definition.