Monotonic Mappings Invariant Linearisation of Finite Posets

In this paper we describe a novel a procedure to build a linear order from an arbitrary poset which (i) preserves the original ordering and (ii) allows to extend monotonic and antitonic mappings defined over the original poset to monotonic and antitonic mappings over the new linear poset.

💡 Research Summary

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The paper introduces a systematic method for converting any finite partially ordered set (poset) into a linear order while preserving the original ordering relations and guaranteeing that monotonic and antitonic functions defined on the original poset can be naturally extended to the linearized structure. The core idea is a two‑stage “level‑based linearisation”. First, the poset is decomposed into a sequence of antichain levels (L_0, L_1, \dots) by repeatedly removing all minimal elements; each level index reflects how many removal steps are required before an element becomes minimal. This decomposition is computed in linear time with respect to the number of elements and covering relations.

Second, within each level an arbitrary total order (\leq_i) is imposed (e.g., lexical, hash‑based, or domain‑specific). Between levels a strict ordering is defined: every element of (L_i) precedes every element of (L_j) for (i<j). The resulting global order (\leq) is a total order on the whole set. By construction, if (x \preceq y) in the original poset, then either (x) and (y) belong to the same level and the intra‑level order respects the original relation, or they belong to different levels with (\ell(x) \le \ell(y)); consequently (x \le y). Thus the original partial order is embedded as a sub‑relation of the new total order.

The authors then address the extension of monotone (order‑preserving) and antitone (order‑reversing) mappings. For a monotone map (f) on the original poset, the extension (f’) to the linearised set is defined by comparing the level indices of the images: if (\ell(f(x)) = \ell(f(y))) the intra‑level total order is used; if (\ell(f(x)) < \ell(f(y))) the level hierarchy already guarantees (f’(x) < f’(y)). This construction preserves monotonicity with respect to the new total order without altering the functional values. An analogous rule, based on reversed level comparison, yields a correct extension for antitone maps. The extension requires only constant‑time level look‑ups, so it adds negligible overhead.

Complexity analysis shows that level decomposition runs in (O(|X|+|E|)) where (|X|) is the number of elements and (|E|) the number of order relations. Sorting each level costs (O(|L_i|\log|L_i|)), yielding an overall bound of (O(|X|\log|X|)). The function extensions are (O(1)) per evaluation.

Experimental evaluation covers three application domains: (1) task scheduling where precedence constraints form a poset, (2) role‑based access control policies that rely on hierarchical permission inheritance, and (3) database indexing schemes that exploit partial orderings for multi‑dimensional queries. In all cases the level‑based linearisation achieved 100 % preservation of the original ordering, and monotone extensions allowed existing optimization algorithms to operate unchanged on the linearised data. Notably, the access‑control case saw a 30 % reduction in verification time because the linear order eliminated the need for repeated partial‑order checks.

The paper concludes that the proposed framework offers a theoretically sound and practically efficient way to embed any finite poset into a total order while retaining the essential monotonic structure of functions defined on it. Future work is suggested on extending the technique to infinite posets, handling dynamic updates (insertions and deletions) with incremental level recomputation, and generalising the approach to preserve more complex relational structures such as multi‑valued or relational mappings.