Algebraic Structure - Hybrid Structures

Hybrid Structures

Algebraic structures can also coexist with added structure of a non-algebraic nature, such as a partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure.

• Topological group: a group with a topology compatible with the group operation.
• Lie group: a topological group with a compatible smooth manifold structure.
• Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order.
• Archimedean group: a linearly ordered group for which the Archimedean property holds.
• Topological vector space: a vector space whose M has a compatible topology.
• Normed vector space: a vector space with a compatible norm. If such a space is topologically complete then it is called a Banach space.
• Hilbert space: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.
• Vertex operator algebra
• Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology.