Hyperalgebras <http://www.algebraic.net/cag/hyperalgebra.html> <http://www.algebraic.net/cag/hyperalgebra.html> <http://www.algebraic.net/cag/hyperalgebra.html> http://www.algebraic.net/cag/hyperalgebra.html Zhaohua Luo (5/2008) Part I Abstract: A hyperalgebra is an algebra of type (0, 0, ..., 2, 3, 4, ...) satisfying three axioms. Finitary hyperalgebras form a coreflective full subcategory of the variety of hyperalgebras, which is equivalent to the opposite of the category of varieties. Thus any subvariety of the variety of hyperalgebras may be viewed as a hypervariety, i.e. a variety of varieties in the sense of W. D. Neumann. Definition. A hyperalgebra is a nonempty set A together with a sequence X = {x_1, x_2, ...} of elements of A and a sequence S = {s_1, s_2, ...} of operations s_n: A^{n+1} -> A, which satisfies the following axioms for M, M_1, ..., M_m, N_1, ..., N_n in A: Write M[M_1, ..., M_n] for s_n(M, M_1, ..., M_n). A1. x_n[M_1, ..., M_m] = M_n if n < m + 1. A2. (M[M_1, ..., M_m])[N_1, ..., N_n] = M[M_1[N_1, ..., N_n], ..., M_m[N_1, ..., N_n]]. A3. M[M_1, ..., M_m] = M[M_1, ..., M_m, M_m]. A hyperalgebra A is finitary if for any M in A there is n > 0 such that M = M[x_1, ..., x_n]. Let A be a hyperalgebra. A model of A is a set D together with a sequence U = {u_1, u_2,...} of operations u_n: A x D^n -> D, which satisfying the following axioms for M, M_1, ..., M_m in A and a_1, ..., a_n in D: Write M[a_1, ..., a_n] for u_n(M, a_1, ..., a_n). B1. x_n[a_1, ..., a_m] = a_n if n < m + 1. B2. (M[M_1, ..., M_m])[a_1, ..., a_n] = M[M_1[a_1, ..., a_n], ..., M_m[a_1, ..., a_n]]. B3. M[a_1, ..., a_n] = M[a_1, ..., a_n, a_n]. Define homomorphisms of models in an obvious way. Denote by Mod(A) the category of models of A. Theorem. 1. If V is a variety and T(V) is the free algebra of V over X, then T(V) is naturally a finitary hyperalgebra, and V is equivalent to Mod(T(V)) as concrete categories over Set . 2. If A is a hyperalgebra then the class Mod(A) forms a variety. If A is a finitary hyperalgebra then it is isomorphic to T(Mod(A)). 3. The correspondences A -> Mod(A) and V -> T(V) establish an equivalence between the category of finitary hyperalgebras and the opposite of the category of varieties.