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The full space is a polyhedral cone.

The zero-space \( \{ 0\} \subseteq N \) is a polyhedral cone.

Two polyhedral cones are combinatorially equivalent if there is an order isomorphism between their face lattices.

A pointed cone \( \sigma \subseteq N \) is called finitely generated, if \( \sigma = \operatorname{Cone}(S) \) for some finite set \( S \subseteq N \).

For any subset \( S \subseteq N \) and \( u \in S^{\vee } \), we have

For a set \( S \subseteq N \), the cone generated by \( S \) is the pointed cone

For a set \(S \subseteq N\), the convex hull of \(S\) is

If \( \sigma , \tau \subseteq N \) are polyhedral, then the convex hull of the two is polyhedral too.

If \( \sigma , \tau \subseteq N \) are polyhedral, the direct sum \( \sigma \oplus _p \tau \) is polyhedral too.

For a finite set \( S \subseteq N \), we have \( S^{\vee \vee } = \operatorname{Cone}(S) \).

If \( \sigma \subseteq N \) is polyhedral, then \( \sigma ^{\vee \vee } = \sigma \).

For a set \( S \subseteq M \), the dual cone of \( S \) is the pointed cone

Face lattices of dual cones are dual to each other.

If \( \sigma \subset N \) is finitely generated, then \( \sigma ^\vee \subset M \) is polyhedral.

If \( \sigma \subseteq N \) is polyhedral, then \( \sigma ^\vee \subseteq M \) is polyhedral too.

A face of a pointed cone \( \sigma \subseteq N \) is the intersection of \( \sigma \) with some hyperplane \( u^{\perp } \) for \( u \in \sigma ^{\vee } \). We write \( \tau \preceq \sigma \) if \( \tau \) is a face of \( \sigma \) and \( \tau \prec \sigma \) if \( \tau \) is a proper face, i.e. \( \tau \preceq \sigma \) and \( \tau \neq \sigma \).

Let \( \tau \preceq \sigma \) be a face of a pointed cone. Then \( \tau ^{\vee } = \tau ^{\perp } + \sigma ^{\vee } \).

A face of a face of a polyhedral cone \( \sigma \) is again a face of \( \sigma \).

The face lattice of a face is isomorphic to a lower interval of the face lattice of a polyhedral cone.

If \( \sigma \) is a pointed cone, then the intersection of two faces of \( \sigma \) is a again face of \( \sigma \).

Given a polyhedral cone, the partial order on its faces is a lattice. We call it the face lattice of the cone.

Given a polyhedral cone, the partial order on its faces is graded.

A face of a polyhedral cone is polyhedral.

A polyhedral cone has only finitely many faces.

If \( \sigma \subseteq N \) is a polyhedral cone and \( w \in N \), then \( \sigma + \operatorname{Cone}(\{ w\} ) \) is polyhedral too.

Any halfspace is a polyhedral cone.

If \( \sigma , \tau \subseteq N \) are polyhedral, then \( \sigma \cap \tau \) is polyhedral too.

If \( \sigma , \tau \subseteq N \) are polyhedral, their join is polyhedral too.

Any linear subspace is a polyhedral cone.

If \( \sigma \subseteq N \) is polyhedral and \(f : N -{\gt} M\) is a linear map, then \( f.map \sigma \subseteq M \) is polyhedral too.

A hyperplane is the perp space of a point.

A halfspace is the dual of a point. The dimension of the module the cone is a submodule of.

The dimension of a pointed cone.

A polyhedral cone is full-dimensional if its dimension is the same as the dimension of the ambient space.

The set of all \(k\)-dimensional faces of a polyheadral cone.

For each \(k\) gives the number of \(k\)-dimensional faces of a polyheadral cone.

Linear transformaions preserve combinatorial equivalence. There is an iso that should be provided

For any subset \( A \subseteq M \), we define the perp space

which is a pointed cone in \( N \). We also write \( u^{\perp } := \{ u\} ^{\perp } \).

If \( A \subseteq M \) is closed under negation (for instance, \( A \subseteq M \) a \( \Bbbk \)-submodule), then \( A^{\perp } = A^{\vee } \).

A pointed cone \( \sigma \subseteq N \) is a \( \Bbbk _{\geq 0} \)-submodule of \( N \).

A pointy cone \( \sigma \subseteq N \) is a \( \Bbbk _{\geq 0} \)-submodule of \( N \) that does not contain a nontrivial linear space.

Every polyhedral cone can be written as the minkowski sum of a pointy cone and a linear subspace.

For a pointed cone \( \sigma \), the polar cone of \( \sigma \) is the pointed cone

Polyhedral cones for a lattice under intersections and convex hulls (its the same lattice as on pointed cones).

Another word for finitely generated cone with a finitely generated canonical dual.

A pointed cone \( \sigma \subseteq N \) is finitely generated iff it is polyhedral.

If \( \sigma \in N \) is finitely generated, then it is polyhedral.

A polytope is a set that can be written as \(\operatorname{Conv}(S)\) for some finite set \(S\).

If \( \sigma , \tau \subseteq N \) are polyhedral, then the cartesian product \( \sigma \prod \tau \) is polyhedral too.

A simplicial cone \(\sigma \) is a polyhedral cone that can be generated by \(dim (\sigma )\) many generators.

If \( \sigma , \tau \subseteq N \) are polyhedral, then \( \sigma + \tau \) is polyhedral too.