Metadata

bibkey: wang2025semigrouprings
authors: Guoqing Wang
year: 2025
title: Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups
doi: null
url: https://arxiv.org/abs/2509.18724v1
claim: Conjecture 3.26 asserts that finite algebraic Davenport constant forces a uniform finite bound on relative constants in the Green quotient over nongroup cyclic elements; Conjecture 5.3 asserts a separate finite-semigroup formula using relative constants and Schutzenberger translation groups.
strata_touched:
  - D5/S0/Certificates/Algebra/WangConjecture326Refutation
  - D5/S0/Certificates/Algebra/WangConjecture53Refutation
license: citation-only
triage: anchor

Algebraic independence relative to semigroup idempotents

Guoqing Wang, Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups, arXiv:2509.18724v1, submitted 2025-09-23. The versioned TeX source Semigroupring.tex has SHA256 e7623bab3697c424a07377677775b2af336352a22a9f5e8e666b750fabde6bf6. The source archive has SHA256 f413479d65c2eeffcbd50ed2468fd2ba558f8033decade5767314848e5b12c1c.

Verified locator

URL: https://arxiv.org/abs/2509.18724v1
Versioned source: v1, submitted 2025-09-23.
Source locator: Section 5, Conjecture 5.3, TeX lines 842–851 and HTML anchor S5.Thmtheorem3.
Definition scope: Section 2, especially lines 203–213 for the ordinary semigroup algebra and Definition 2.1; Section 5, lines 842–851 for the complete conjecture hypotheses, equality, and epsilon definition.

Conjecture 3.26 is the complete periodic-semigroup implication addressed by the infinite construction below. Section 2 supplies the positive cyclic and periodic conventions, principal ideals and Green relation, ordinary semigroup algebra, Definition 2.1 and its independently nonzero coefficients, and sequence conventions with Definition D’s relative Davenport constant. The note binds the separate 3.26 and 5.3 formal modules listed above; each construction addresses only its own conjecture.

Source statement

Conjecture 5.3, TeX line 842, HTML anchor S5.Thmtheorem3, states:

Let S be a finite commutative semigroup, and let K be a field. Suppose that either (i) K is a splitting field of S, or (ii) Char(K)=p and exp(S) is a p-power. Then d(S,K)=max over a in S of {epsilon_a D_bar(a)(S/H)+d(Gamma(H_a),K)}, where epsilon_a is 0 if the positive cyclic semigroup generated by a is a group and 1 otherwise.

Section 2, TeX lines 203–213, uses the ordinary semigroup algebra: all finitely supported coefficient functions on S with convolution. An absorbing semigroup element remains an independent basis vector. Definition 2.1 requires the product of X^s_i-a_i X^e(s_i) to be nonzero for every tuple of individually nonzero scalars. The constant is a supremum of lengths. The formal model takes the supremum of finite lengths in ENat; infinity means unbounded finite lengths, with no infinite-product interpretation.

The positive cyclic semigroup contains positive multiples only. Its unique idempotent is e(a); its period is the least positive eventual shift of the positive multiples. The exponent is the LCM of periods, not of indices or cyclic-semigroup sizes. Lines 245–246 define splitting by exactly exp(S) solutions of zeta^exp(S)=1 in K. Branch (i) imposes no algebraic-closure or characteristic-zero condition.

Green H means equality of the principal ideals {a} union {a+c:c in S}. The quotient is the congruence quotient with its induced operation. Definition D, line 239, defines the relative Davenport constant at a by irreducible sequences summing to a, with the empty proper subsequence included and D_identity=0. It is not the global reducibility threshold. The Schutzenberger carrier consists of actual translation maps on H_a realized by stabilizing elements of S, identifying equal maps. No identity is adjoined. The paper assigns d(empty,K)=0.

Six-label mechanism

Take S={0,1} x {0,1,2}, with

$(b, n) + (c, m) = (max (b, c), min (2, n + m)),$

and K=Q. This is a finite commutative monoid, an admissible instance of the source’s semigroup quantifier. Every cyclic period is one and Q splits exponent one. The identity is (0,0). The cyclic idempotent of (b,0) is (b,0); that of (b,1) or (b,2) is (b,2).

For t=(0,1), the coefficient of X^t in X^t-a X^(0,2) is one for every rational a. Conversely, choosing every coefficient to be one kills every sequence of length at least two: idempotent labels give zero factors, and for any two transient labels x,y the four products x+y, x+e(y), e(x)+y, e(x)+e(y) coincide. Thus d(S,Q)=1, uniformly over all finite lengths.

Principal ideals distinguish all six elements, so the actual H quotient retains their distinctions. At a=(1,1), the quotient sequence with terms (0,1) and (1,0) sums to a, while its proper subsequences sum to (0,0), (0,1), and (1,0). It is irreducible and its relative Davenport constant is at least two. The positive cyclic semigroup generated by a is not a group, so epsilon_a=1. Since the remaining summand is nonnegative, the conjectured right side is at least two. Its exact maximum is unnecessary.

Scope and provenance

The Conjecture 5.3 assertion quantifies over finite commutative semigroups, including those without an identity, and over all fields satisfying either source branch. Its counterexample is an ordinary rational semigroup algebra; no contracted algebra or replacement for the relative constant is involved. Positive products avoid a nonunital empty-product assumption. Adjoining length zero to a supremum of natural lengths does not affect its value. Empty sequence sums refer only to an identity already present in the semigroup. The rational witness avoids zero-ring conventions.

The source and bounded later-literature screen supplied with issue 23871 found no exact settlement in the inspected scope. This is not an exhaustive absence or priority claim. The construction’s novelty is not established. Conjecture 5.3 and Conjecture 3.26 are separate assertions. The six-label argument above concerns only 5.3; the infinite construction below concerns 3.26.

Conjecture 3.26: one infinite periodic semigroup

Conjecture 3.26 states, verbatim:

Let S be a commutative periodic semigroup, and let R be a commutative unitary ring. If d(S,R) < infinity then sup {D_bar(a)(S/H): a in S and is not a group} < infinity.

Take the single carrier S=Finset(N) x {0,1,2}, with

$(A, k) + (B, l) = (A \cup B, min (2, k + l)),$

and R=Q. The map i maps to ({i},0) is injective, so S is infinite. For every x=(A,k), put E(x)=(A,0) if k=0 and E(x)=(A,2) otherwise. The positive cyclic subsemigroup is exactly {x,E(x)}. Thus every cyclic subsemigroup is finite and E(x) is its unique idempotent. This is pointwise periodicity, without imposing finiteness on S.

Choosing coefficient one at every position makes every product of at least two factors zero. Idempotent terms have a zero factor. For two level-one terms x,y, all four basis products in (X^x-X^E(x))(X^y-X^E(y)) have the same label (A union B,2), so their coefficients cancel. Ordinary convolution, including the retained basis label at level two, is essential. This proves the sufficient bound d(S,Q)<=1; no exact value is needed for Conjecture 3.26.

The principal ideal of (A,k) consists exactly of (B,l) with A subset B and k<=l. Mutual ideal membership forces equality of both coordinates, so the actual Green quotient map is injective on all of this infinite S.

For every n, let a_n=({0,…,n-1},1). Use the sequence with one term (empty,1) and one term ({i},0) for each i<n. It has n+1 terms and sum a_n. The weight w(A,k)=|A|+1(k!=0) is subadditive. Every term has weight one, while w(a_n)=n+1. Every subsequence with that sum must therefore have at least n+1 terms. This proves irreducibility, including against the empty subsequence. Injectivity of the actual quotient map transfers this result to S/H, giving D_bar(a_n)(S/H)>=n+1 for every n in the same S.

The positive cyclic semigroup of a_n has precisely the level-one and level-two labels. Neither acts as an identity on a_n. It is not a group, even when its identity could differ from the ambient monoid identity. Consequently the source’s filtered supremum is infinite. This refutes Conjecture 3.26 as one complete implication; no finite truncation or varying-semigroup family is used. An exact formula for each relative constant, a minimal counterexample, and a novelty or priority claim are not asserted. The later-literature screen is bounded by the source and preregistration 23896.

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