View source for Heaps and semi-heaps

{{TEX|done}}{{MSC|20N10}}

Algebraic structures 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 [[#References|[1]]]. Heaps have been studied from their abstract aspect [[#References|[2]]], [[#References|[3]]]. In particular, it was shown by R. Baer [[#References|[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|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 [[#References|[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|differential geometry]] in the study of coordinate atlases [[#References|[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, [[#References|[4]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Prüfer,   "Theorie der Abelschen Gruppen"  ''Math. Z.'' , '''20'''  (1924)  pp. 165–187</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R. Baer,   "Zur Einführung des Scharbegriffs"  ''J. Reine Angew. Math.'' , '''160'''  (1929)  pp. 199–207</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  J. Certaine,   "The ternary operation $(abc)=ab^{-1}c$ of a group"  ''Bull. Amer. Math. Soc.'' , '''49'''  (1943)  pp. 869–877</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Vagner,   "The theory of generalized heaps and of generalized groups"  ''Mat. Sb.'' , '''32''' :  3  (1953)  pp. 545–632  (In Russian)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  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)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  M. Flato,   "Deformation view of physical theories"  ''Czechoslovak J. Phys.'' , '''B32'''  (1982)  pp. 472–475</TD></TR>
</table>

====Comments====
The terms ''groud'', ''abstract coset'', ''restricted brigade'', ''Schar'' and ''flock'' are also found in the literature.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> Bruck, Richard Hubert ''A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. '''20''' Springer (1958) {{ZBL|0081.01704}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> Hollings, Christopher ''Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups'', Amer. Math. Soc. (2014) {{ZBL|1317.20001}}  </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> Schein, Boris "Inverse semigroups and generalised groups". In A.F. Lavrik. Twelve papers in logic and algebra. Amer. Math. Soc. Transl. 113. American Mathematical Society. pp. 89–182. ISBN 0-8218-3063-5 {{ZBL|0247.20060}} {{ZBL|0404.20055}}  </TD></TR>
</table>