Heaps and semi-heaps

Algebras with one ternary operation that satisfies certain identities. Heaps are defined by the identities

$$[[x_1x_2x_3]x_4x_5]=[x_1x_2[x_3x_4x_5]],$$

$$[x_1x_1x_2]=x_2,\quad[x_1x_2x_2]=x_1,$$

while semi-heaps are defined by the identities

$$[[x_1x_2x_3]x_4x_5]=[x_1[x_4x_3x_2]x_5]=[x_1x_2[x_3x_4x_5]].$$

All heaps are also semi-heaps.

If, in the set $\Phi(A,B)$ of all one-to-one mappings of a set $A$ into a set $B$, one defines the ternary operation that puts an ordered triplet of mappings $\phi_1,\phi_2,\phi_3$ into correspondence with the mapping that is the composite of $\phi_1,\phi_2^{-1},\phi_3$, then $\Phi(A,B)$ is a heap. Any heap is isomorphic to some heap of one-to-one mappings. If a ternary operation is introduced into an arbitrary group $G$ by putting $[g_1g_2g_3]=g_1g_2^{-1}g_3$, a heap is obtained (the heap associated with the given group). The concept of a heap was introduced in the study of the above ternary operation on an Abelian group [1]. Heaps have been studied from their abstract aspect [2], [3]. In particular, it was shown by R. Baer [2] that if an arbitrary given element $s_0$ is fixed in a heap $S$, then the operations defined by the equations $s_1s_2=[s_1s_0s_2]$, $s^{-1}=[s_0ss_0]$ define a group structure on $S$ in which $s_0$ is the unit; the heap associated with this group coincides with the initial heap, while the groups obtained from a heap by fixing various elements of it are isomorphic. In other words, the variety of all heaps is equivalent to the variety of all groups.

The set $\mathfrak P(A,B)$ of all binary relations (cf. Binary relation) between the elements of two sets $A$ and $B$ is a semi-heap with respect to the triple multiplication $[\rho_1\rho_2\rho_3]=\rho_1\rho_2^{-1}\rho_3$. The set of all invertible partial mappings of $A$ into $B$ is also closed with respect to the triple multiplication and is a generalized heap [4], i.e. a semi-heap with the identities

$$[xxx]=x,$$

$$[x_1x_1[x_2x_2x_3]]=[x_2x_2[x_1x_1x_3]],$$

$$[[x_1x_2x_2]x_3x_3]=[[x_1x_3x_3]x_2x_2].$$

Generalized heaps find application in the foundations of differential geometry in the study of coordinate atlases [5]. Heaps are closely connected with semi-groups with involution. If an involution $\theta$, which is an anti-automorphism, is defined on a semi-group $S$, then the ternary operation $[s_1s_2s_3]=s_1\theta(s_2)s_3$ converts $S$ into a semi-heap. Any semi-heap is isomorphic to a sub-semi-heap of a semi-group with involution, [4].

References

[1]	H. Prüfer, "Theorie der Abelschen Gruppen" Math. Z. , 20 (1924) pp. 165–187
[2]	R. Baer, "Zur Einführung des Scharbegriffs" J. Reine Angew. Math. , 160 (1929) pp. 199–207
[3]	J. Certaine, "The ternary operation $(abc)=ab^{-1}c$ of a group" Bull. Amer. Math. Soc. , 49 (1943) pp. 869–877
[4]	V.V. Vagner, "The theory of generalized heaps and of generalized groups" Mat. Sb. , 32 : 3 (1953) pp. 545–632 (In Russian)
[5]	V.V. Vagner, "Foundations of differential geometry and contemporary algebra" , Proc. 4-th All-Union. Mat. Conf. , 1 , Leningrad (1963) pp. 17–29 (In Russian)
[6]	M. Flato, "Deformation view of physical theories" Czechoslovak J. Phys. , B32 (1982) pp. 472–475