Hypergraphs-and-Orthoalgebras

A repository of the programs used in my summer project about hypergraphs and orthoalgebras. In particular, the programs are mostly for displaying graphs of certain algebras, and for checking validity of candidate orthoalgebras.

It is necessary to have Graphviz installed in order to display the graphs. Info on Graphviz can be found here: https://graphviz.readthedocs.io/en/stable/manual.html.

Table of Contents

• Displaying Hasse diagrams
  • Powesets-ps_hasse
  • Orthoalgebras (and Boolean algebras) - oa_hasse
• Checking Orthoalgebras - oa_check
• Generating Orthoalgebras - oa_gen
• Displaying Hypergraphs - hg_show

Displaying Hasse diagrams

Powersets - ps_hasse

This program allows you to generate the Hasse diagram of a powerset. For example:

list = powerset([1,2]) # returns [[1,2],[1],[2],[]]
ps_hasse(list, 2) # returns Hasse diagram of list

Returns

Orthoalgebras (and Boolean Algebras) - oa_hasse

This program allows you to generate the Hasse diagram of an orthoalgebra (or a Boolean algebras). For example:

A note about the datatypes NEG, COMP, and OPLUS. Suppose the orthoalgebra in question contains elements.

• NEG is an -element list such that it contains only the elements .
• COMP is a symmetric matrix with each element either or . If COMP[i,j] == 1, then elements and are compatible.
• OPLUS is a symmetric matrix with entries satisfying
  • if COMP[i,j] == 1, then OPLUS[i,j] , where is the partial binary operation in the orthoalgebra
  • if COMP[i,j] == 0, then OPLUS[i,j] = -1, i.e. an element not in the orthoalgebra.

Checking Orthoalgebras - oa_check

This program allows you to check if an algebra (in the datatype defined above) is an orthoalgebra (or a Boolean algebra) or not. It does this by checking the axioms in the definition an orthoalgebra.

Examples:

A valid orthoalgebra.

An invalid orthoalgebra.

Generating Orthoalgebras - oa_gen

The program generates orthoalgebras by searching for an OPLUS matrix (if any) which satisfies the orthoalgebra axioms and corresponds to a given COMP matrix.

Explanation of functions:

• oa_check2 checks if OPLUS satisfies the orthoalgebra associativity axiom
• ass_checker checks if certains in OPLUS are associative with all other entries non-negative entries
• generator checks if a valid OPLUS matrix exists for a specific COMP matrix
• random_comp creates a random COMP matrix
• temp_oplus creates a template OPLUS matrix with all necessary entries pre-filled
• neg creates a valid NEG array of a given length

Examples:

Finding a valid orthoalgebra.

Determining no solution exists.

Displaying Hypergraphs - hg_show

This program display hypergraphs.

Explanation of datatypes:

• ATOMS is a list of strings (which correspond points in the graph)
• LINES is a list containing 3-element lists (which correspond to the lines in the graph)
• PLANES is a list containing 7-element lists (which correspond to the planes in the graph)

Examples:

A point and a line.

A plane.