A construction of all homomorphisms of a heterogeneous algebra into an algebra of the same type is presented. A relational structure is assigned to any heterogeneous algebra, and homomorphisms between these relational structures make it possible to construct homomorphisms between heterogeneous algebras. Homomorphisms of relational structures can be constructed using homomorphisms of algebras that are described in [11].
The main result of this paper is the introduction of a notion of a generalized RLatin square, which includes as a special case the standard Latin square, as well as the magic square, and also the double stochastic matrix. Further, the algebra of all generalized Latin squares over a commutative ring with identity is investigated. Moreover, some remarkable examples are added.
Let G be a group and H an abelian group. Let J ∗ (G, H) be the set of solutions f : G → H of the Jensen functional equation f(xy) + f(xy−1 ) = 2f(x) satisfying the condition f(xyz) − f(xzy) = f(yz) − f(zy) for all x, y,z ∈ G. Let Q ∗ (G, H) be the set of solutions f : G → H of the quadratic equation f(xy) + f(xy−1 ) = 2f(x) + 2f(y) satisfying the Kannappan condition f(xyz) = f(xzy) for all x,y, z ∈ G. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution f : G → H of the Whitehead equation is of the form 4f = 2ϕ + 2ψ, where 2ϕ ∈ J ∗ (G, H) and 2ψ ∈ Q ∗ (G, H). Moreover, if H has the additional property that 2h = 0 implies h = 0 for all h ∈ H, then every solution f : G → H of the Whitehead equation is of the form 2f = ϕ+ψ, where ϕ ∈ J ∗ (G, H) and 2ψ(x) = B(x,x) for some symmetric bihomomorphism B : G × G → H.
States on commutative basic algebras were considered in the literature as generalizations of states on MV-algebras. It was a natural question if states exist also on basic algebras which are not commutative. We answer this question in the positive and give several examples of such basic algebras and their states. We prove elementary properties of states on basic algebras. Moreover, we introduce the concept of a state-morphism and characterize it among states. For basic algebras which are the certain pastings of Boolean algebras the construction of a state-morphism is shown.