gh-37930: Work on `CircuitsMatroid`
    
Various class-specific improvements. Implementation of enumerators (e.g.
`in/dependent_r_sets`) and `_closure`.
(The methods `bases`, `nonbases`, `circuit_closures`, `flats`, etc., are
now much faster.)

The changes in `matroid.pyx` are mostly for consistency's sake
(enumerators returning a `SetSystem`).
    
URL: #37930
Reported by: gmou3
Reviewer(s): gmou3, Travis Scrimshaw

Author: Release Manager

src/sage/algebras/orlik_solomon.py

@@ -132,7 +132,7 @@ def __init__(self, R, M, ordering=None):
             self._broken_circuits[frozenset(L[1:])] = L[0]
 
         cat = Algebras(R).FiniteDimensional().WithBasis().Graded()
-        CombinatorialFreeModule.__init__(self, R, M.no_broken_circuits_sets(ordering),
+        CombinatorialFreeModule.__init__(self, R, list(M.no_broken_circuits_sets(ordering)),
                                          prefix='OS', bracket='{',
                                          sorting_key=self._sort_key,
                                          category=cat)

src/sage/algebras/orlik_terao.py

@@ -25,7 +25,7 @@ class OrlikTeraoAlgebra(CombinatorialFreeModule):
     r"""
     An Orlik-Terao algebra.
 
-    Let `R` be a commutative ring. Let `M` be a matroid with ground set
+    Let `R` be a commutative ring. Let `M` be a matroid with groundset
     `X` with some fixed ordering and representation `A = (a_x)_{x \in X}`
     (so `a_x` is a (column) vector). Let `C(M)` denote the set of circuits
     of `M`. Let `P` denote the quotient algebra `R[e_x \mid x \in X] /
@@ -65,7 +65,7 @@ class OrlikTeraoAlgebra(CombinatorialFreeModule):
 
     - ``R`` -- the base ring
     - ``M`` -- the defining matroid
-    - ``ordering`` -- (optional) an ordering of the ground set
+    - ``ordering`` -- (optional) an ordering of the groundset
 
     EXAMPLES:
 
@@ -164,7 +164,7 @@ def __init__(self, R, M, ordering=None):
             self._broken_circuits[frozenset(L[1:])] = L[0]
 
         cat = Algebras(R).FiniteDimensional().Commutative().WithBasis().Graded()
-        CombinatorialFreeModule.__init__(self, R, M.no_broken_circuits_sets(ordering),
+        CombinatorialFreeModule.__init__(self, R, list(M.no_broken_circuits_sets(ordering)),
                                          prefix='OT', bracket='{',
                                          sorting_key=self._sort_key,
                                          category=cat)
@@ -252,7 +252,7 @@ def algebra_generators(self):
         r"""
         Return the algebra generators of ``self``.
 
-        These form a family indexed by the ground set `X` of `M`. For
+        These form a family indexed by the groundset `X` of `M`. For
         each `x \in X`, the `x`-th element is `e_x`.
 
         EXAMPLES::
@@ -323,7 +323,7 @@ def product_on_basis(self, a, b):
         TESTS:
 
         Let us check that `e_{s_1} e_{s_2} \cdots e_{s_k} = e_S` for any
-        subset `S = \{ s_1 < s_2 < \cdots < s_k \}` of the ground set::
+        subset `S = \{ s_1 < s_2 < \cdots < s_k \}` of the groundset::
 
             sage: # needs sage.graphs
             sage: G = Graph([[1,2],[1,2],[2,3],[3,4],[4,2]], multiedges=True)
@@ -355,11 +355,11 @@ def product_on_basis(self, a, b):
     def subset_image(self, S):
         r"""
         Return the element `e_S` of ``self`` corresponding to a
-        subset ``S`` of the ground set of the defining matroid.
+        subset ``S`` of the groundset of the defining matroid.
 
         INPUT:
 
-        - ``S`` -- a frozenset which is a subset of the ground set of `M`
+        - ``S`` -- a frozenset which is a subset of the groundset of `M`
 
         EXAMPLES::
 

src/sage/matroids/circuits_matroid.pxd

@@ -2,23 +2,30 @@ from sage.matroids.matroid cimport Matroid
 from sage.matroids.set_system cimport SetSystem
 
 cdef class CircuitsMatroid(Matroid):
-    cdef frozenset _groundset  # _E
-    cdef int _matroid_rank  # _R
-    cdef SetSystem _C  # circuits
+    cdef frozenset _groundset
+    cdef int _matroid_rank
+    cdef set _C  # circuits
     cdef dict _k_C  # k-circuits (k=len)
+    cdef list _sorted_C_lens
     cdef bint _nsc_defined
     cpdef groundset(self)
     cpdef _rank(self, X)
     cpdef full_rank(self)
-    cpdef _is_independent(self, F)
-    cpdef _max_independent(self, F)
-    cpdef _circuit(self, F)
+    cpdef _is_independent(self, X)
+    cpdef _max_independent(self, X)
+    cpdef _circuit(self, X)
+    cpdef _closure(self, X)
 
     # enumeration
     cpdef bases(self)
+    cpdef nonbases(self)
+    cpdef independent_r_sets(self, long r)
+    cpdef dependent_r_sets(self, long r)
     cpdef circuits(self, k=*)
     cpdef nonspanning_circuits(self)
-    cpdef no_broken_circuits_sets(self, ordering=*)
+    cpdef no_broken_circuits_facets(self, ordering=*, reduced=*)
+    cpdef no_broken_circuits_sets(self, ordering=*, reduced=*)
+    cpdef broken_circuit_complex(self, ordering=*, reduced=*)
 
     # properties
     cpdef girth(self)