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Algebras with one ternary operation that satisfies certain identities. Heaps are defined by the identities
 
Algebras with one ternary operation that satisfies certain identities. Heaps are defined by the identities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467601.png" /></td> </tr></table>
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$$[[x_1x_2x_3]x_4x_5]=[x_1x_2[x_3x_4x_5]],$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467602.png" /></td> </tr></table>
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$$[x_1x_1x_2]=x_2,\quad[x_1x_2x_2]=x_1,$$
  
 
while semi-heaps are defined by the identities
 
while semi-heaps are defined by the identities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467603.png" /></td> </tr></table>
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$$[[x_1x_2x_3]x_4x_5]=[x_1[x_4x_3x_2]x_5]=[x_1x_2[x_3x_4x_5]].$$
  
 
All heaps are also semi-heaps.
 
All heaps are also semi-heaps.
  
If, in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467604.png" /> of all one-to-one mappings of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467605.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467606.png" />, one defines the ternary operation that puts an ordered triplet of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467607.png" /> into correspondence with the mapping that is the composite of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467608.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h0467609.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676010.png" /> by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676011.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676012.png" /> is fixed in a heap <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676013.png" />, then the operations defined by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676015.png" /> define a group structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676016.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676017.png" /> 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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676018.png" /> of all binary relations (cf. [[Binary relation|Binary relation]]) between the elements of two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676020.png" /> is a semi-heap with respect to the triple multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676021.png" />. The set of all invertible partial mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676023.png" /> is also closed with respect to the triple multiplication and is a generalized heap [[#References|[4]]], i.e. a semi-heap with the identities
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676024.png" /></td> </tr></table>
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$$[xxx]=x,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676025.png" /></td> </tr></table>
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$$[x_1x_1[x_2x_2x_3]]=[x_2x_2[x_1x_1x_3]],$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676026.png" /></td> </tr></table>
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$$[[x_1x_2x_2]x_3x_3]=[[x_1x_3x_3]x_2x_2].$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676027.png" />, which is an anti-automorphism, is defined on a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676028.png" />, then the ternary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676029.png" /> converts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676030.png" /> into a semi-heap. Any semi-heap is isomorphic to a sub-semi-heap of a semi-group with involution, [[#References|[4]]].
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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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046760/h04676031.png" /> 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>
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<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>

Revision as of 10:50, 15 November 2014

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
How to Cite This Entry:
Heaps and semi-heaps. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Heaps_and_semi-heaps&oldid=13359
This article was adapted from an original article by V.N. Salii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article