Zero Cancellation and Equation Structure in Kiselman's Semigroup

Abstract: We investigate equations in Kiselman's semigroup $K_n$, generated by $a_1, \dots, a_n$. Let $f$ denote the zero element of $K_n$. We prove that if $y \in K_n$ lies in the subsemigroup generated by $a_2, \dots, a_n$, then $x y = f$ implies $x = f$. In contrast, the equation $x a_1 = f$ admits non-trivial solutions. We describe the solution set of this equation, show that its cardinality is $1 + |K_{n-1}|$, and study its algebraic structure. Moreover, we show that $|K_{2n+1}|$ is even, whereas $|K_{2n}|$ is odd.

• The paper establishes the Zero Cancellation Theorem, showing that when an equation in Kiselman’s semigroup yields the unique zero, a factor from a specific generator subset forces the other factor to be zero.
• It employs canonical word forms and rewriting techniques to rigorously characterize the structure of solutions, notably for equations like x a₁ = zero.
• The study delivers combinatorial parity results for |Kₙ| and offers insights relevant to algorithmic equation solving and symbolic algebra in non-group semigroups.

Zero Cancellation and Equation Structure in Kiselman's Semigroup

Introduction and Context

The paper "Zero Cancellation and Equation Structure in Kiselman's Semigroup" (2604.22007) presents a comprehensive investigation into the behavior of equations involving the zero element in the Kiselman semigroup $K_n$. The Kiselman semigroup, defined via generators $a_1,\dots,a_n$ and relations combining idempotency and certain absorption-like rules, encapsulates a combinatorial algebraic structure with connections to operator semigroups in convex analysis and graph dynamical systems. Prior research has established normal forms, basis problems, combinatorics, and finiteness results for Kiselman and Hecke–Kiselman monoids. This work exploits canonical word forms and deep structural properties to analyze when certain equations involving the zero element necessarily collapse, the nature of cancellation laws, and to enumerate the solution sets to particular equations.

Zero Cancellation Theorem

The central result, termed the Zero Cancellation Theorem, establishes strong restrictions on solutions to equations of the form $xy = f$, where $f$ is the unique zero in $K_n$. Specifically:

• If $y$ is in the subsemigroup generated by $a_2,\dots,a_n$, then $x y = f$ implies $x = f$.
• Conversely, if $x$ is generated by $a_1,\dots,a_n$0, $a_1,\dots,a_n$1 implies $a_1,\dots,a_n$2.

This result sharply delineates the propagation of the zero element in terms of the "content" map—i.e., which generators appear in the canonical form of an element. The proof is structured using canonical forms and rewriting relations, particularly noting the unique role of $a_1,\dots,a_n$3 in disrupting zero cancellation. A key technical insight is that if zero is achieved by "appending" from the subsemigroup generated without $a_1,\dots,a_n$4, this can only occur if the initial factor is itself zero, barring the exceptional influence of $a_1,\dots,a_n$5. The result is further supported by corollaries specifying that for $a_1,\dots,a_n$6, the only solution to $a_1,\dots,a_n$7 is $a_1,\dots,a_n$8, and analogously on the left.

Structure of Solutions to $a_1,\dots,a_n$9

Distinct from the general zero cancellation scenario, the equation $xy = f$0 admits a rich solution structure. The paper precisely describes the set $xy = f$1:

• $xy = f$2 (strong quantitative result).
• $xy = f$3 decomposes as the disjoint union of $xy = f$4 and a set $xy = f$5 of elements in $xy = f$6 (those canonical forms containing $xy = f$7) that satisfy $xy = f$8.

The set $xy = f$9 is shown to be in bijection with $f$0 via a projection onto the part before $f$1 in the canonical representative. The explicit algebraic description of $f$2 is given as

$f$3

where $f$4 depends on minimality with respect to becoming zero by right multiplication. An explicit multiplication table within $f$5 is proven, showing it forms a subsemigroup: it essentially behaves as a Bruck–Reilly extension with absorbing zero. The analysis leverages intricate word combinatorics and normal form uniqueness.

Canonical Forms and Structural Properties

A significant portion of the paper systematically develops and uses canonical forms for elements of $f$6. These canonical words have unique minimal-length representatives and enjoy transitive closure rewriting properties that mirror standard techniques from term rewriting systems. The interplay between right and left deletion operations in the rewriting system, controlled by the semigroup's defining relations, is critical to the technical apparatus. The content map further allows for transferring information about idempotents, automorphisms, and combinatorial invariants into the solution structure for zero equations.

Parity of $f$7 and Combinatorial Implications

The paper concludes with an application of the developed techniques to the combinatorial enumeration of $f$8:

• $f$9 is always even, whereas $K_n$0 is always odd.

This is derived using a recursive decomposition of $K_n$1 based on whether $K_n$2 or $K_n$3 (or both/neither) appear in the canonical form, and by constructing involutive bijections between certain classes of elements using a content-reversing map $K_n$4. The result invokes symmetry properties and canonical form combinatorics. Such parity results are nontrivial and relevant to the combinatorial landscape of semigroups, impacting enumeration, representation theory, and potentially the analysis of random walks or statistical models built upon $K_n$5.

Implications and Prospects

On the theoretical side, the zero cancellation phenomena sharpen our understanding of how semigroup identities propagate in Kiselman-type structures and provide blueprints for analogous results in other diagrammatic or relation-based semigroups—especially those with a unique zero or absorbing element. The fine structure of solution sets to nontrivial zero equations underlines the subtlety introduced by minimal generators such as $K_n$6, exposing the limits of classical cancellation theory in non-group semigroups.

Practically, this can guide algorithmic approaches to the word problem and equation solving in semigroups with zero, which is relevant for symbolic algebra implementations concerned with relations and idempotency. The parity results indicate further combinatorial or enumerative phenomena possibly tied to symmetry classes or the structure of canonical word languages.

Future developments may include generalizations to broader classes of Hecke–Kiselman structures, investigations into their automorphisms in dynamical settings, analysis of their representation theory, and applications to combinatorial optimization and theoretical computer science contexts where such semigroups serve as algebraic models for process dynamics, rewriting, or resource consumption.

Conclusion

This paper provides rigorous analysis of equation structure in Kiselman's semigroup, delineating sharp conditions for zero cancellation, quantifying and describing the solution space to $K_n$7, and elucidating the parity of the order of $K_n$8. The methods showcase the power of canonical form-based reasoning within algebraic combinatorics, and the results contribute both to the algebraic theory of semigroups and to the toolkit for investigating equations and identities in operator-generated algebraic systems.