Order-Preserving Decreasing Transformations

• Order-preserving decreasing transformations are mappings that reduce each element’s value while maintaining the given order, defined by fixed-point and isometry conditions.
• Their structure is classified within transformation semigroups using Green's relations and explicit combinatorial enumeration, such as binomial coefficients.
• This framework yields practical insights in representation theory, automata, and combinatorics by linking algebraic constraints with counting formulas and structural properties.

Order-Preserving Decreasing Transformations encompass a rich class of algebraic, combinatorial, and analytic mappings, primarily studied in the framework of transformation semigroups and operator theory. In the algebraic setting, such transformations typically act on finite chains or ordered structures and are defined either as partial isometries that move points only downward, or as order-preserving maps that decrease each element relative to its original position. In functional analysis, the notion often generalizes to operators between convex (or operator) functions that reverse or preserve order, especially within duality theory. The following sections provide a comprehensive account based on results found in (Kehinde et al., 2011), with references to related lines of research.

1. Algebraic Definition and Cycle Structure

Order-preserving decreasing transformations are formalized in the context of the symmetric inverse semigroup $\mathcal{I}_n$ over the finite chain $X_n = \{1,2,\ldots,n\}$. The subsemigroups $\mathcal{DDP}_n$ and $\mathcal{ODDP}_n$ respectively consist of order-decreasing partial isometries (those that are distance-preserving and satisfy $a(x) \le x$ for all $x$ in their domain) and the further restriction to order-preserving maps.

Key cycle-theoretic properties include:

• Isometry Condition: For every $x, y$ in the domain, $|x - y| = |a(x) - a(y)|$.
• Order-Decreasing Condition: $a(x) \le x$ for all $x$.
• Fixed Points: The set $F(a) = \{x \in X_n : a(x) = x\}$ characterizes idempotency; it is shown that $f(a) \in \{0, 1, h(a)\}$, where $h(a) = |\operatorname{Im} a|$.
• For $\mathcal{ODDP}_n$, the drop $x - a(x)$ is constant across the domain: for $a \in \mathcal{ODDP}_n$, $x - a(x) = r$ for some fixed $r$, so $a(x) = x - r$ (Lemma 1.10).
• Idempotency and Cycles: Maps with $|F(a)|>1$ must be partial identities (Corollary 1.4).

This rigid structure simplifies the classification and enumeration of such maps.

2. Green's Relations and Semigroup Structure

Green's relations provide a canonical partition of semigroups. In $\mathcal{DDP}_n$ and $\mathcal{ODDP}_n$, it is established (Theorem 2.1) that both are J-trivial semigroups, meaning the J-relation is trivial (only the diagonal), indicating no nontrivial "reachability" via ideal-generated elements.

The starred analogues further refine:

• $a ~\mathcal{R}^*~ b$ iff $\mathrm{Dom}\ a \subseteq \mathrm{Dom}\ b$.
• $a \le^{+} b$ iff $\mathrm{Im}\ a \subseteq \mathrm{Im}\ b$.
• $a ~\mathcal{H}^*~ b$ iff both domain and image coincide.

These equivalence classes are completely characterized by the domains and images, and in the order-preserving case, by the uniform shift $r$.

3. 0-E-Unitary Ample Structure

The semigroup $\mathcal{ODDP}_n$ is further shown to be a $0$-E-unitary ample semigroup (Theorem 2.6). In this context:

• If $e$ is a nonzero idempotent and $e \cdot s$ is idempotent (and nonzero), then $s$ must itself be an idempotent.
• The ample property ensures compatibility between idempotents and the semigroup multiplication, with explicit relations:
  • $e \cdot a = a \cdot (e \cdot a)^*$ and $a \cdot e = (a \cdot e)^{+} \cdot a$, where $(e \cdot a)^*, (a \cdot e)^{+}$ are unique idempotents in corresponding Green's classes.

These structural constraints guarantee a "clean" algebraic interplay between map composition, order-decreasing behavior, and idempotency.

4. Enumerative Combinatorics and Equivalence Class Cardinalities

The cycle and semigroup structures enable combinatorial enumeration:

• Elements by Height: $F(n; p) = |\{ a \in S : h(a) = p \}|$.
  • For $S = \mathcal{ODDP}_n$, $F(n; p) = \binom{n+1}{p+1}$ for $1 \le p \le n$ (Prop. 3.3).
  • Total order: $$|\mathcal{ODDP}_n| = \sum_{p=0}^n \binom{n+1}{p+1} = 2^{n+1} - (n+1)$$ (Thm. 3.4).
• Elements by Number of Fixed Points: Similar enumeration for $F(n; m) = |\{ a : f(a) = m \}|$.
• For $\mathcal{DDP}_n$: Additional orders and recursions are derived, e.g., the number of transformations with no fixed points relates to smaller semigroups: $F(n; 0) = |U_{n-1}|$.

These formulas yield exact counts and reveal new combinatorial sequences, some not previously recorded.

5. Interplay with Related Semigroups and Algebraic Implications

Order-preserving decreasing transformations are closely related to, and often form subsemigroups of, other well-studied transformation semigroups:

• The symmetric inverse semigroup, the Catalan monoid (for full order-preserving, order-decreasing maps), and the Schröder monoid (for partial order-preserving, order-decreasing maps) share key properties or embeddings.
• Presentation theory, as explored in (Umar, 2017), generalizes these results, showing that defining relations for these semigroups are shaped by idempotency, commutation, and elementary "reduction" steps.

The identification of similar combinatorial and algebraic frameworks facilitates transfer of techniques across structures, underscores the recurring role of binomial and Catalan-type numbers, and informs broader theory of transformation semigroups and automata.

6. Summary of Key Formulas

Central enumerative and structural formulas pertinent to $\mathcal{ODDP}_n$ are as follows: $$h(a) = |\operatorname{Im} a|;\quad F(a) = \{ x \in X_n : a(x) = x \};\quad f(a) = |F(a)|$$ Constant drop property for $a \in \mathcal{ODDP}_n$: $$\forall x, y \in \operatorname{Dom} a,\quad x - a(x) = y - a(y)$$ Cardinality: $F(n; p) = \binom{n+1}{p+1}$

$$| \operatorname{ODDP}_n | = \sum_{p=0}^n \binom{n+1}{p+1} = 2^{n+1} - (n+1)$$

7. Broader Implications and Research Directions

The investigation of order-preserving decreasing transformations underpins advances in:

• Representation Theory: Ideal structures and regularity properties inform the construction and analysis of representations for these semigroups.
• Combinatorics: The emergence of new triangles of numbers (sequences) and their combinatorial interpretation enrich existing enumerative frameworks.
• Automata and Formal Language Theory: Because order-preserving decreasing maps frequently appear within the context of automata on ordered languages and piecewise testable properties, their algebraic characterization aids both theoretical analysis and algorithmic development.

Furthermore, the results suggest that imposing both isometry and order-decreasing conditions on finite chains results in highly constrained algebraic and combinatorial structures, amenable to explicit description and enumeration (Kehinde et al., 2011).

Table: Cardinal Characterizations in $\mathcal{ODDP}_n$

Statistic	Formula	Range
Elements of height $p$	$F(n; p) = \binom{n+1}{p+1}$	$1 \le p \le n$
Total elements	$2^{n+1} - (n+1)$	$n \geq 1$
Fixed points possibilities	$f(a) \in \{0, 1, h(a)\}$	For $a$ in $\mathcal{ODDP}_n$

These results illustrate the explicit enumerability and strong combinatorial regularity inherent to this class of semigroups.

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Order-preserving decreasing transformations are thus characterized by strong algebraic constraints, detailed combinatorial behavior, trivial or fine-tuned Green’s relations, and enumeration tied to binomial number patterns. The paper of their semigroup structure, regular elements, and equivalence classes is deeply interconnected and foundational in both algebraic and combinatorial theory.