Implement algorithm for enumerating ideals of a poset in O(log n) amortized time

src/doc/en/reference/references/index.rst

@@ -5881,6 +5881,10 @@ REFERENCES:
             Journal of Algorithms, Volume 26, Issue 2, Pages 275 - 290, 1998.
             (https://doi.org/10.1006/jagm.1997.0891)
 
+.. [Sq2002] Matthew B. Squire.
+            *Enumerating the Ideals of a Poset*.
+            (https://citeseer.ist.psu.edu/465417.html)
+
 .. [SS1983] Shorey and Stewart. "On the Diophantine equation a
             x^{2t} + b x^t y + c y^2 = d and pure powers in recurrence
             sequences." Mathematica Scandinavica, 1983.

src/sage/categories/posets.py

@@ -305,6 +305,56 @@ def order_filter(self, elements):
                 [3, 7, 8, 9, 10, 11, 12, 13, 14, 15]
             """
 
+        def generate_upsets(U, Y, Poset):
+            r"""
+            Enumerates the ideals (upsets) of a given poset.
+
+            ALGORITHM:
+
+            The algorithm is based on [Sq2002]_.
+            Current algorithms for genertating ideals require an amortized
+            time of `O(n)` per ideal in the worst case, where `n` is the number
+            of elements in the poset. This algorithm generates ideals in an
+            amortized time of `O(log n)` per ideal.
+
+            The function recursively generates all possible upsets (subsets) of
+            a given set `Y` based on a partially ordered set (Poset), ensuring that
+            if `Xi < Xm < Xj`, then Xi must also be less than `Xj`. It iterates through
+            `Y`, partitioning it into two subsets based on the relationship between
+            elements and yields the resulting upsets.
+
+            INPUT:
+
+            - ``U`` -- a list of the current upset being constructed.
+            - ``Y`` -- a list of remaining elements to be considered for inclusion.
+            - ``Poset`` -- a dictionary representing the partial order.
+
+            OUTPUT:
+
+            - A subset of Y that satisfies the upset condition.
+
+            EXAMPLES::
+
+            """
+            if not Y:
+                yield U
+            else:
+                Xm = Y[len(Y) // 2]
+                Y1 = []
+                Y2 = []
+                for Xi in Y:
+                    if Poset[(Xi, Xm)] == 1:
+                        U[Y.index(Xi)] = 0
+                    else:
+                        Y1.append(Xi)
+                yield from Posets.generate_upsets(U.copy(), Y1, Poset)
+                for Xj in Y:
+                    if Poset[(Xm, Xj)] == 1:
+                        U[Y.index(Xj)] = 1
+                    else:
+                        Y2.append(Xj)
+                yield from Posets.generate_upsets(U.copy(), Y2, Poset)
+
         def directed_subset(self, elements, direction):
             r"""
             Return the order filter or the order ideal generated by a