Hyper Pseudo Effect Algebras

Definition: A partial binary hyper operation on A is a partial mapping from A×A into P(A)^*, where P(A)^* is the set of all non-empty subsets of A.

We say that A + B is defined, iff for any a ∈ A and b ∈ B is a + b defined. The outcome of A + B is then

⋃_a∈A,b∈B (a + b).

We say that A + a is defined, iff A + {a} is defined and A + a = ⋃_u∈A (u + a).

We say that a + A is defined, iff {a} + A is defined and a + A = ⋃_u∈A (a + u).

Definition: The set A equipped with partial hyper operation +, unary operations ^–, ^~ and constants 0, 1 is said to be a hyper pseudo effect algebra, if the following properties are satisfied:

1. Operation + is partially associative, i.e. x + y is defined and (x + y) + z is defined, iff y + z is defined and x + (y + z) is defined and in such case (x + y) + z = x + (y + z).
2. For any x∈A there are unique y∈A and z∈A such that 1 ∈ x + y and 1 ∈ z + x. (y is denoted as x^– and z as x^~.)
3. x + 0 and 0 + x are defined for any x in A.
4. 1 + x is defined, iff x = 0; x + 1 is defined, iff x = 0.
5. Relation ≤ defined as: (x≤y, iff x + y^– is defined) is reflexive and antisymmetric.
   Relation can be equivalently defined as (x≤y, iff y^~ + x is defined).

If moreover ≤ is transitive, we say that A is a transitive hyper pseudo effect algebra..

Finite models

Case - parameters to hyper_pseudo_EA, where m.n is entered as m n (space separated).

# generated - models generated by hyper_pseudo_EA program

# of models - excluding isomorphic models by iso_hpea

n = 2

n = 3

n = 4

n = 5

Source code (26.7 kB)