gh-35240: ABC for `BooleanPolynomialRing`
    
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### 📚 Description
The new ABC `sage.rings.polynomial.multi_polynomial_ring_base.BooleanPol
ynomialRing_base` can be imported for `isinstance` tests, without a
compile time dependency on `brial`.

This is a step towards a modularized distribution **sagemath-brial**.

Part of:
- #32414

### 📝 Checklist

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- [x] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies
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URL: #35240
Reported by: Matthias Köppe
Reviewer(s): François Bissey, Matthias Köppe

Author: Release Manager

src/sage/rings/polynomial/multi_polynomial_ring_base.pxd

@@ -9,3 +9,7 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
     cdef public dict _magma_cache
 
     cdef _coerce_c_impl(self, x)
+
+
+cdef class BooleanPolynomialRing_base(MPolynomialRing_base):
+    pass

src/sage/rings/polynomial/multi_polynomial_ring_base.pyx

@@ -1750,6 +1750,28 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
         return DifferentialWeylAlgebra(self)
 
 
+cdef class BooleanPolynomialRing_base(MPolynomialRing_base):
+    r"""
+    Abstract base class for :class:`~sage.rings.polynomial.pbori.pbori.BooleanPolynomialRing`.
+
+    This class is defined for the purpose of ``isinstance`` tests.  It should not be
+    instantiated.
+
+    EXAMPLES::
+
+        sage: from sage.rings.polynomial.multi_polynomial_ring_base import BooleanPolynomialRing_base
+        sage: R.<x, y, z> = BooleanPolynomialRing()
+        sage: isinstance(R, BooleanPolynomialRing_base)
+        True
+
+    By design, there is only one direct implementation subclass::
+
+        sage: len(BooleanPolynomialRing_base.__subclasses__()) <= 1
+        True
+    """
+    pass
+
+
 ####################
 # Leave *all* old versions!
 

src/sage/rings/polynomial/multi_polynomial_sequence.py

@@ -1257,14 +1257,16 @@ def eliminate_linear_variables(self, maxlength=Infinity, skip=None, return_reduc
 
             This is called "massaging" in [BCJ2007]_.
         """
-        from sage.rings.polynomial.pbori.pbori import BooleanPolynomialRing,gauss_on_polys
-        from sage.rings.polynomial.pbori.ll import eliminate,ll_encode,ll_red_nf_redsb
+        from sage.rings.polynomial.multi_polynomial_ring_base import BooleanPolynomialRing_base
 
         R = self.ring()
 
-        if not isinstance(R, BooleanPolynomialRing):
+        if not isinstance(R, BooleanPolynomialRing_base):
             raise NotImplementedError("Only BooleanPolynomialRing's are supported.")
 
+        from sage.rings.polynomial.pbori.pbori import gauss_on_polys
+        from sage.rings.polynomial.pbori.ll import eliminate, ll_encode, ll_red_nf_redsb
+
         F = self
         reductors = []
 
@@ -1340,10 +1342,11 @@ def _groebner_strategy(self):
             sage: F._groebner_strategy()
             <sage.rings.polynomial.pbori.pbori.GroebnerStrategy object at 0x...>
         """
-        from sage.rings.polynomial.pbori.pbori import BooleanPolynomialRing
+        from sage.rings.polynomial.multi_polynomial_ring_base import BooleanPolynomialRing_base
+
         R = self.ring()
 
-        if not isinstance(R, BooleanPolynomialRing):
+        if not isinstance(R, BooleanPolynomialRing_base):
             from sage.libs.singular.groebner_strategy import GroebnerStrategy
             return GroebnerStrategy(self.ideal())
         else:
@@ -1452,7 +1455,6 @@ def solve(self, algorithm='polybori', n=1,  eliminate_linear_variables=True, ver
             []
 
         """
-        from sage.rings.polynomial.pbori.pbori import BooleanPolynomialRing
         from sage.modules.free_module import VectorSpace
 
         S = self
@@ -1462,6 +1464,8 @@ def solve(self, algorithm='polybori', n=1,  eliminate_linear_variables=True, ver
         if eliminate_linear_variables:
             T, reductors = self.eliminate_linear_variables(return_reductors=True)
             if T.variables() != ():
+                from sage.rings.polynomial.pbori.pbori import BooleanPolynomialRing
+
                 R_solving = BooleanPolynomialRing( T.nvariables(), [str(_) for _ in list(T.variables())] )
             S = PolynomialSequence( R_solving, [ R_solving(f) for f in T] )
 
@@ -1543,11 +1547,13 @@ def reduced(self):
             ....:             assert g[i].lt() not in t.divisors()
         """
 
-        from sage.rings.polynomial.pbori.pbori import BooleanPolynomialRing
+        from sage.rings.polynomial.multi_polynomial_ring_base import BooleanPolynomialRing_base
+
         R = self.ring()
 
-        if isinstance(R, BooleanPolynomialRing):
+        if isinstance(R, BooleanPolynomialRing_base):
             from sage.rings.polynomial.pbori.interred import interred as inter_red
+
             l = [p for p in self if not p==0]
             l = sorted(inter_red(l, completely=True), reverse=True)
             return PolynomialSequence(l, R, immutable=True)

src/sage/rings/polynomial/pbori/pbori.pxd

@@ -1,14 +1,13 @@
 from libcpp.memory cimport unique_ptr, shared_ptr, make_shared
 
-from sage.rings.polynomial.multi_polynomial_ring_base cimport \
-                                                MPolynomialRing_base
+from sage.rings.polynomial.multi_polynomial_ring_base cimport MPolynomialRing_base, BooleanPolynomialRing_base
 from sage.rings.polynomial.multi_polynomial cimport MPolynomial
 from sage.structure.element cimport MonoidElement
 
 from sage.libs.polybori.decl cimport *
 
 
-cdef class BooleanPolynomialRing(MPolynomialRing_base):
+cdef class BooleanPolynomialRing(BooleanPolynomialRing_base):
     cdef PBRing _pbring
     cdef Py_ssize_t* pbind
     cdef public _monom_monoid

src/sage/rings/polynomial/pbori/pbori.pyx

@@ -256,7 +256,7 @@ block_dp_asc = int(pbblock_dp_asc)
 rings = sage.misc.weak_dict.WeakValueDictionary()
 
 
-cdef class BooleanPolynomialRing(MPolynomialRing_base):
+cdef class BooleanPolynomialRing(BooleanPolynomialRing_base):
     """
     Construct a boolean polynomial ring with the following parameters:
 
@@ -409,7 +409,7 @@ cdef class BooleanPolynomialRing(MPolynomialRing_base):
             pbnames = tuple(names)
             names = [name.replace('(', '').replace(')', '') for name in pbnames]
 
-        MPolynomialRing_base.__init__(self, GF((2,1)), n, names, order)
+        BooleanPolynomialRing_base.__init__(self, GF((2,1)), n, names, order)
 
         counter = 0
         for i in range(len(order.blocks()) - 1):
@@ -7615,7 +7615,7 @@ cdef BooleanPolynomialRing BooleanPolynomialRing_from_PBRing(PBRing _ring):
 
     self._pbring = _ring
 
-    MPolynomialRing_base.__init__(self, GF(2), n, names, T)
+    BooleanPolynomialRing_base.__init__(self, GF(2), n, names, T)
 
     self._zero_element = new_BP(self)
     (<BooleanPolynomial>self._zero_element)._pbpoly = PBBoolePolynomial(0, self._pbring)