Two-Sided Wreath Product in Semigroup Theory

• Two-Sided Wreath Product is defined via λρ-systems with left and right extractors ensuring associativity through specific coherence laws.
• It recovers the ordinary wreath product by varying coordinate sets and relaxing symmetry, thereby extending classical semidirect constructions.
• The framework refines the Krohn–Rhodes theorem using categorical and universal constructions, with implications for semigroup and quantum algebra.

In semigroup theory, the two-sided wreath product is treated through the $\lambda\rho$-product, a construction built from a semigroup $\mathbf S=(S,\cdot)$, a family of sets $I[a]$ indexed by $a\in S$, and structure maps $\lambda[a,b]\colon I[ab]\to I[a]$ and $\rho[a,b]\colon I[ab]\to I[b]$. For any semigroup $\mathbf H=(H,\cdot)$, one forms the disjoint union

$$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$

with multiplication

$$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$

The central fact is that this multiplication is associative if and only if the data satisfy three coherence laws. In this usage, “two-sided” refers to the simultaneous presence of a left extractor $\lambda$ and a right extractor $\mathbf S=(S,\cdot)$0, rather than to the ordinary semidirect-product wreath product with a single coordinate action (Botur et al., 2018, Botur et al., 15 Sep 2025).

1. $\mathbf S=(S,\cdot)$1-systems as the semigroup-theoretic definition

A $\mathbf S=(S,\cdot)$2-system over $\mathbf S=(S,\cdot)$3 consists of the sets $\mathbf S=(S,\cdot)$4 and maps $\mathbf S=(S,\cdot)$5, $\mathbf S=(S,\cdot)$6 satisfying

$\mathbf S=(S,\cdot)$7

$\mathbf S=(S,\cdot)$8

$\mathbf S=(S,\cdot)$9

These identities are the exact compatibility conditions needed for associativity of $I[a]$0. The 2018 paper presents this as the semigroup construction “related to the two-sided wreath product,” while the 2025 paper formulates the same equivalence as Theorem 1: $I[a]$1 is a semigroup for every semigroup $I[a]$2 if and only if $I[a]$3 is a $I[a]$4-system (Botur et al., 2018, Botur et al., 15 Sep 2025).

The semigroup-theoretic motivation is that the classical wreath-product picture is too rigid for arbitrary semigroups. The $I[a]$5-formalism allows the coordinate set to vary with the multiplier and does not require bijections everywhere. The 2018 paper states that this is the right two-sided analogue of the classical wreath product because it drops unnecessary symmetry assumptions and keeps exactly the data needed for semigroup multiplication to be associative (Botur et al., 2018).

2. Recovery of ordinary wreath products and the classical two-sided case

The ordinary wreath product appears as a special case when all fibers are equal to a fixed set $I[a]$6. If $I[a]$7 acts on $I[a]$8 on the right, one sets

$I[a]$9

Then

$a\in S$0

which is exactly the usual wreath product $a\in S$1. The 2025 paper goes further: its Theorem 4 states that a $a\in S$2-system is group-preserving if and only if its skeleton is a group and the system is unital, if and only if it is isomorphic to one coming from a genuine group action on a set. In that case the action on $a\in S$3 is recovered by

$a\in S$4

so in the group case the generalized construction becomes the ordinary wreath product up to isomorphism (Botur et al., 15 Sep 2025).

The specifically semigroup-theoretic two-sided wreath product arises from a two-sided action on a set $a\in S$5. The required identities are

$a\in S$6

With

$a\in S$7

the associated $a\in S$8-product is the two-sided wreath product. The 2018 paper explicitly states that if $a\in S$9 is a two-sided action of a semigroup on a set, then the corresponding $\lambda[a,b]\colon I[ab]\to I[a]$0-system produces a semigroup isomorphic to the two-sided wreath product (Botur et al., 2018).

3. Unitality, categorical organization, and universal constructions

A $\lambda[a,b]\colon I[ab]\to I[a]$1-system is unital precisely when the skeleton $\lambda[a,b]\colon I[ab]\to I[a]$2 is a monoid and

$\lambda[a,b]\colon I[ab]\to I[a]$3

for all $\lambda[a,b]\colon I[ab]\to I[a]$4. The 2025 paper states that this is exactly the condition under which $\lambda[a,b]\colon I[ab]\to I[a]$5 is a monoid whenever $\lambda[a,b]\colon I[ab]\to I[a]$6 is a monoid. In the unital setting one can reduce the data to single-parameter maps $\lambda[a,b]\colon I[ab]\to I[a]$7 and $\lambda[a,b]\colon I[ab]\to I[a]$8, leading to pre-$\lambda[a,b]\colon I[ab]\to I[a]$9-systems and their reconstruction under a natural-solutions condition. The same paper also records the functorial formula

$\rho[a,b]\colon I[ab]\to I[b]$0

showing that compatible transformations of systems induce homomorphisms of the associated products (Botur et al., 15 Sep 2025).

The 2018 paper packages the theory categorically. For a fixed semigroup $\rho[a,b]\colon I[ab]\to I[b]$1, the category $\rho[a,b]\colon I[ab]\to I[b]$2 has $\rho[a,b]\colon I[ab]\to I[b]$3-systems as objects and slice transformations $\rho[a,b]\colon I[ab]\to I[b]$4 with $\rho[a,b]\colon I[ab]\to I[b]$5 as arrows. This yields a contravariant functor

$\rho[a,b]\colon I[ab]\to I[b]$6

and the Grothendieck construction then produces a category of general $\rho[a,b]\colon I[ab]\to I[b]$7-systems in which the skeleton semigroup itself may vary. The same paper also constructs a free or universal unital $\rho[a,b]\colon I[ab]\to I[b]$8-system over a free monoid $\rho[a,b]\colon I[ab]\to I[b]$9: for a word $\mathbf H=(H,\cdot)$0, the set $\mathbf H=(H,\cdot)$1 consists of sequences

$\mathbf H=(H,\cdot)$2

satisfying

$\mathbf H=(H,\cdot)$3

for each adjacent pair. Every $\mathbf H=(H,\cdot)$4-product divides a product arising from this free system, giving the construction an explicit universal-algebraic meaning (Botur et al., 2018).

4. Degenerate cases, examples, and the Krohn–Rhodes refinement

Several specializations recover standard semigroup constructions. If $\mathbf H=(H,\cdot)$5 is the trivial semigroup, then $\mathbf H=(H,\cdot)$6. The same conclusion holds when every fiber $\mathbf H=(H,\cdot)$7 is empty. If every $\mathbf H=(H,\cdot)$8, then $\mathbf H=(H,\cdot)$9. The 2025 paper adds that if

$$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$0

then $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$1 is a two-sided ideal of $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$2. It also states that the construction naturally encompasses block products, and that Example 5 produces a 5-element semigroup from a two-element base that cannot be a nontrivial semidirect product (Botur et al., 15 Sep 2025).

A principal application is a refinement of the Krohn–Rhodes theorem. Using a $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$3-system over the two-element semilattice $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$4, the 2018 paper shows that $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$5 has a quotient isomorphic to the left flip-flop monoid. The resulting corollary is: $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$6 This is presented as a slightly finer decomposition, with the three-element flip-flop replaced by the two-element semilattice (Botur et al., 2018).

5. Terminological variation in adjacent literatures

Outside semigroup theory, many papers use the word “wreath product” for the standard semidirect-product construction and explicitly state that no formal two-sided wreath product is being introduced. This is true for imprimitive wreath products of symmetric and alternating groups, for wreath products embedded in automorphism groups of two-sided full shifts, and for the wreath-product model of “spinning switches” puzzles (East et al., 2021, Salo, 2020, Kagey, 2022).

Context	Construction	Status of “two-sided”
Symmetric and alternating groups	$$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$7 with coordinates permuted on the right	Standard permutational wreath product
Cellular automata on full shifts	$$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$8 under an $$H^{[\mathcal S]}=\biguplus_{a\in S} H^{I[a]}=\{(x,a):a\in S,\ x\in H^{I[a]}\},$$9-ithful action	Restricted wreath product inside a two-sided shift setting
Spinning switches	$$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$0 with $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$1	Standard semidirect-product wreath product
Block permutations of symmetric groups	$$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$2	Block-permutation realization of ordinary wreath-product multiplication
Association schemes over a poset	Generalized wreath product over $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$3	No separate named two-sided wreath product

Block-permutation and association-scheme usages sharpen this ambiguity. The 2018 paper on $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$4 studies what its summary calls the “two-sided wreath product of symmetric groups,” but the group is explicitly $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$5 with the ordinary multiplication

$$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$6

By contrast, the 2023 paper on association schemes states that it does not define a separate “two-sided wreath product”; instead, its generalized wreath product over a finite poset $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$7 unifies the usual wreath product and the direct product, with two-sided behavior appearing only implicitly through poset closures and two-sided ideals in adjacency and Terwilliger algebras (Tout, 2018, Watanabe, 2023).

This suggests that the phrase “two-sided wreath product” is strongly context-dependent. In semigroup theory it denotes a specific left/right transport mechanism encoded by $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$8 and $$(x,a)\star (y,b)=\bigl((x\circ\lambda[a,b])\cdot (y\circ\rho[a,b]),\,ab\bigr).$$9; in neighboring literatures it may instead refer to block structure, to two input actions, or merely to surrounding two-sided algebraic phenomena.

6. Higher-level and quantum analogues

Further wreath-type constructions appear in representation theory and quantum algebra. The 2026 paper on higher-level affine wreath product algebras defines the higher-level affine wreath product category $\lambda$0 and the algebra

$\lambda$1

where $\lambda$2 is the number of black strands and $\lambda$3 is the number of red strands. When $\lambda$4, one recovers the usual affine wreath product algebra: $\lambda$5 The paper explicitly states that it does not define a genuinely two-sided wreath product algebra; the closest analogues are left/right sliding relations, a reversed monoidal supercategory, and two-sided ideals in cyclotomic quotients (Moran, 5 May 2026).

In compact quantum groups, the analogous construction is the free wreath product. The 2016 paper does not define a separate two-sided wreath product, but its final generalized definition takes two faithful, centrally ergodic, Markov trace-preserving actions $\lambda$6 and $\lambda$7 and defines a free wreath action $\lambda$8 characterized by

$\lambda$9

The paper states that this is the closest thing to a two-sided wreath product in that setting, because both factors are treated on equal footing as action-data (Tarrago et al., 2016).

These developments suggest that the semigroup-theoretic two-sided wreath product is one member of a broader family of wreath-type constructions. What distinguishes it is that the bidirectional structure is built directly into the multiplication law through the pair of coordinate maps $\mathbf S=(S,\cdot)$00 and $\mathbf S=(S,\cdot)$01, rather than being inherited from a single action, a diagrammatic calculus, or a poset-indexed product.