Hom-Heaps: Ternary Structures in Hom-Algebra

• Hom-heaps are ternary algebraic structures defined by a bijective twisting map and a ternary operation satisfying Hom-Mal’tsev and Hom-associativity laws.
• They generalize classical heaps by extending the group structure to a Hom-algebra setting and serve as the additive layer for Hom-trusses and Hom-braces.
• Hom-heap theory integrates concepts like retracts, subheaps, and modules, offering distinct examples that differentiate them from data-structure heaps.

Hom-heaps are ternary algebraic structures introduced as natural Hom-type analogues of classical heaps, with the latter understood as the affine or zero-free form of group structure. In the formulation of “On Hom-Analogues of Heaps and Trusses” (Anowar et al., 1 Sep 2025), a Hom-heap consists of a set $H$, a bijective twisting map $\alpha:H\to H$, and a ternary operation $\langle-,-,-\rangle$ satisfying twisted Mal’tsev and associativity laws. They extend the classical construction $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$ to the Hom-algebraic setting and supply the additive or affine layer for Hom-trusses and, indirectly, Hom-braces (Anowar et al., 1 Sep 2025). In adjacent literature, the same phrase may also refer to heap structures carried by Hom-sets of homotopy classes in unpointed stable homotopy theory, so the term is not entirely uniform across subfields (Vokřínek, 2013).

1. Classical background and conceptual scope

A classical heap is a non-empty set equipped with a ternary operation satisfying associativity together with the Mal’tsev identities

$\langle a,a,b\rangle=b=\langle b,a,a\rangle,$

or, equivalently in the older para-associative formulation, all para-associativity identities plus the degeneracy conditions $[x,x,y]=y$ and $[x,y,y]=x$ (Elhamdadi et al., 2019). Every group yields such a structure by

$\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$

and, conversely, choosing a base point $e$ in a heap recovers a group law

$a+_e b=\langle a,e,b\rangle.$

This is the standard sense in which heaps are “groups without a chosen zero” or “a group without a zero object” (Vokřínek, 2013, Küng, 2024).

Hom-heaps retain this affine viewpoint but replace ordinary associativity and unit-like identities by Hom-twisted analogues. They therefore belong to the same conceptual family as Hom-groups, Hom-trusses, and Hom-braces, where a self-map $\alpha:H\to H$0 controls the deformation of classical algebraic laws (Anowar et al., 1 Sep 2025). This places Hom-heaps within Hom-algebra rather than within the unrelated data-structural literature on priority queues.

2. Definition and basic identities

The basic definition fixes a set $\alpha:H\to H$1, a bijective map $\alpha:H\to H$2, and a ternary operation

$\alpha:H\to H$3

These data form a Hom-heap when the following axioms hold for all $\alpha:H\to H$4: $\alpha:H\to H$5

$\alpha:H\to H$6

and

$\alpha:H\to H$7

The first law is the Hom-Mal’tsev property, the second is the Hom-associativity property, and the third expresses compatibility with the twisting map (Anowar et al., 1 Sep 2025).

When $\alpha:H\to H$8, these axioms reduce to the classical heap axioms. Hom-heaps are therefore a genuine extension of ordinary heap theory rather than an unrelated ternary construction. The paper also singles out the involutive case: a Hom-heap $\alpha:H\to H$9 is involutive when

$\langle-,-,-\rangle$0

Because $\langle-,-,-\rangle$1 is assumed bijective, the theory repeatedly uses $\langle-,-,-\rangle$2. A particularly useful reformulation of Hom-associativity is

$\langle-,-,-\rangle$3

This identity functions as an “untwisted-looking” associativity law and is central in the verification of retract and inverse formulas (Anowar et al., 1 Sep 2025).

3. Correspondence with Hom-groups

The main structural theorem is a Hom-analogue of the classical heap-group correspondence. Starting from a Hom-group $\langle-,-,-\rangle$4, the ternary operation

$\langle-,-,-\rangle$5

defines a Hom-heap (Anowar et al., 1 Sep 2025). This is the direct Hom-variant of the classical formula $\langle-,-,-\rangle$6, with the middle inverse and third entry corrected by $\langle-,-,-\rangle$7.

Conversely, given a Hom-heap and an element $\langle-,-,-\rangle$8, one can define the retract

$\langle-,-,-\rangle$9

The crucial difference from classical heap theory is that this retract yields a Hom-group precisely when the chosen point is fixed by the twisting map: $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$0 Under that hypothesis, $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$1 becomes a Hom-group, with unit $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$2 and inverse

$\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$3

(Anowar et al., 1 Sep 2025).

This fixed-point condition is the decisive new phenomenon. In the classical case, every chosen base point determines a group structure. In the Hom-setting, that is no longer true. The paper emphasizes the obstruction by examples such as $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$4, where generally $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$5; in such a case the chosen retract point cannot serve as a Hom-group identity (Anowar et al., 1 Sep 2025). A common misconception is therefore to treat Hom-heaps as if they were classical heaps with a harmless endomorphism appended. The fixed-point restriction shows that the deformation affects the affine-to-binary passage in an essential way.

4. Subheaps, normality, morphisms, and examples

A subset $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$6 is a Hom-subheap when it is closed under the ternary operation and stable under $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$7: $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$8 A morphism $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c$9 of Hom-heaps preserves both the ternary operation and the twisting maps: $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$0 Bijective morphisms are Hom-heap isomorphisms (Anowar et al., 1 Sep 2025).

Normality is formulated in ternary, not binary, terms. A Hom-subheap $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$1 is normal if there exists $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$2 such that for all $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$3 and $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$4, there exists $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$5 with

$\langle a,a,b\rangle=b=\langle b,a,a\rangle,$6

The paper proves that once this holds for one designated $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$7, it is equivalent to analogous conditions for all $\langle a,a,b\rangle=b=\langle b,a,a\rangle,$8, and to reformulations involving the subsets

$\langle a,a,b\rangle=b=\langle b,a,a\rangle,$9

and the symmetry relation

$[x,x,y]=y$0

(Anowar et al., 1 Sep 2025). This gives normal subheaps a role analogous to normal subgroups, but expressed in the ambient ternary language.

The paper’s examples show both the breadth of the notion and the strictness of the Hom-generalization.

Construction	Twisting map	Feature
$[x,x,y]=y$1 on $[x,x,y]=y$2	$[x,x,y]=y$3	involutive Hom-heap
$[x,x,y]=y$4 from a heap automorphism	automorphism $[x,x,y]=y$5	twisting construction from an ordinary heap
$[x,x,y]=y$6 on $[x,x,y]=y$7	$[x,x,y]=y$8	classical heap viewed as Hom-heap
$[x,x,y]=y$9 on $[x,y,y]=x$0, $[x,y,y]=x$1	$[x,y,y]=x$2	Hom-heap but not a heap

The last example is especially significant: it satisfies the Hom-heap axioms, yet the underlying ternary operation is not associative in the classical heap sense. This shows that Hom-heaps are not merely classical heaps with a relabeling of identities (Anowar et al., 1 Sep 2025).

5. Role in Hom-trusses, Hom-braces, and modules

Hom-heaps are not an isolated notion. In the same paper they serve as the additive or affine substrate for Hom-trusses. Three types of Hom-trusses are introduced. Type $[x,y,y]=x$3 uses an abelian heap with an endomorphism $[x,y,y]=x$4, whereas types $[x,y,y]=x$5 and $[x,y,y]=x$6 use an abelian Hom-heap $[x,y,y]=x$7 together with a second map $[x,y,y]=x$8 and a multiplicative operation $[x,y,y]=x$9 satisfying Hom-associative and distributive laws (Anowar et al., 1 Sep 2025).

For type $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$0, the defining compatibility includes

$\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$1

together with left and right distributivity over the Hom-heap operation: $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$2

$\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$3

Type $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$4 replaces these by more heavily twisted laws involving $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$5, $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$6, and $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$7. A central structural result is that $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$8-Hom-trusses of type $\langle a,b,c\rangle=a\cdot b^{-1}\cdot c,$9 and type $e$0 are equivalent (Anowar et al., 1 Sep 2025).

Hom-braces are developed in parallel, again in three variants. The bridge back to Hom-heaps is the retract construction. When a Hom-truss has multiplicative part a Hom-group, the affine addition is recovered by

$e$1

In this way certain Hom-trusses yield Hom-braces, and conversely certain Hom-braces recover unital Hom-trusses (Anowar et al., 1 Sep 2025). The architecture is therefore layered: Hom-heaps underlie Hom-trusses, and Hom-trusses mediate to Hom-braces.

The module theory follows the same pattern. For an $e$2-Hom-truss of type $e$3,

$e$4

a left $e$5-module is an abelian Hom-heap $e$6 with an action $e$7 satisfying three compatibility identities, including

$e$8

and

$e$9

(Anowar et al., 1 Sep 2025). Hom-heaps thus function as the ambient additive geometry not only for objects but also for representations.

6. Related usages, precursors, and common confusions

The term has a wider background than the single Hom-algebraic definition suggests. In unpointed stable homotopy theory, Vokřínek showed that if $a+_e b=\langle a,e,b\rangle.$0 is a simplicial model category, then the set $a+_e b=\langle a,e,b\rangle.$1 of homotopy classes of maps from an $a+_e b=\langle a,e,b\rangle.$2-dimensional cofibrant object $a+_e b=\langle a,e,b\rangle.$3 to a $a+_e b=\langle a,e,b\rangle.$4-connected fibrant object $a+_e b=\langle a,e,b\rangle.$5 with $a+_e b=\langle a,e,b\rangle.$6 admits a canonical structure of a possibly empty abelian heap (Vokřínek, 2013). In that setting, “Hom-Heaps” refers to heap structures on Hom-sets of homotopy classes rather than to Hom-type twisted heaps.

Küng’s work on Grothendieck heaps extends $a+_e b=\langle a,e,b\rangle.$7 to unpointed Waldhausen categories and unpointed stable $a+_e b=\langle a,e,b\rangle.$8-categories by replacing abelian groups with abelian heaps; it does not define a Hom-heap, but it explicitly treats heaps as the correct zero-free replacement for additive decategorification and cites the appearance of heap structures on morphism spaces as external motivation (Küng, 2024). These developments show that the heap formalism is already important in zero-free homotopical and categorical algebra, independently of Hom-deformation.

A different neighboring direction is the theory of right heaps and left quasiheaps. There the classical right Mal’tsev identity is weakened to a one-sided “right weakly Mal’tsev” law, and pointed right heaps reconstruct right groups rather than groups (Albano et al., 8 Jun 2026). This suggests, though the point is extrapolative, that one-sided Hom-heap theories may arise by twisting asymmetric heap identities rather than the symmetric classical ones.

Ordinary heap cohomology, heap extensions, and categorical heap objects form further untwisted background. Heap cohomology classifies extensions of the form

$a+_e b=\langle a,e,b\rangle.$9

and pointed heap objects in symmetric monoidal categories are characterized as involutory Hopf monoids (Elhamdadi et al., 2019). These results are not themselves Hom-heap theory, but they identify the parts of classical heap theory most likely to admit Hom-analogues.

Finally, the terminology should not be confused with the use of “heap” in data structures. Smooth heaps and hollow heaps are self-adjusting or meldable priority queues, not ternary algebraic or homotopical objects (Kozma et al., 2018, Hansen et al., 2015). The shared word is historical rather than conceptual.