Finite Relation Minimal Maximal Elements
Return all minimal and maximal elements only after validating a supplied finite partial order; these need not be a unique least/greatest element.
Description
Return all minimal and maximal elements only after validating a supplied finite partial order; these need not be a unique least/greatest element.
Minimal and maximal elements of a partial order need not be a single least or greatest element. Several incomparable extremes can exist.
When to use this tool
Return all minimal and maximal elements only after validating a supplied finite partial order; these need not be a unique least/greatest element.
- Prepare the supplied pairsJson, universe fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter pairsJson, universe. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.minimal, result.maximal. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Parse a unique string universe and in-universe pair list. Require reflexivity, antisymmetry and transitivity. An element is minimal if no distinct predecessor pair enters it and maximal if no distinct successor pair leaves it.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| pairsJson | JSON text, entered as one text field. The decoded structure is illustrated below; it is not a separate file. | Text |
| universe | universe. | List of strings; at most 128 entries |
For the JSON text fields, the following decoded structures show the example records before they are serialized into input strings. Keep field names, list nesting and value types as shown.
[
{
"field": "pairsJson",
"decoded": [
[
"a",
"a"
],
[
"b",
"b"
],
[
"a",
"b"
]
]
}
]| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.minimal | The minimal list in the ordering and representation described above. |
| result.maximal | The maximal list in the ordering and representation described above. |
{
"input": {
"pairsJson": "[[\"a\",\"a\"],[\"b\",\"b\"],[\"a\",\"b\"]]",
"universe": [
"a",
"b"
]
},
"output": {
"result": {
"minimal": [
"a"
],
"maximal": [
"b"
]
}
}
}Limitations and assumptions
- Pairs must include the required reflexive and transitive relationships; no closure is added. Output follows universe order and does not claim a unique least or greatest element.
- All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
- JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.
Alternative or Complementary methods
Declare the full universe and distinguish testing a property from adding its closure. Use separate functionality and totality checks before interpreting a relation as a function.
References
Similar or alternative tools
- Finite Relation Partial Order Property Report
Report partial-order property evidence for an explicit finite relation; does not assume a claimed order or repair missing pairs.
- Finite Relation Composition Pairs
Compose finite relations by following first then second, coalescing duplicate witnesses into unique endpoint pairs; order convention is explicit.