# The monoid of generalized hypersubstitutions of type τ = (n)

Wattapong Puninagool; Sorasak Leeratanavalee

Discussiones Mathematicae - General Algebra and Applications (2010)

- Volume: 30, Issue: 2, page 173-191
- ISSN: 1509-9415

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topWattapong Puninagool, and Sorasak Leeratanavalee. "The monoid of generalized hypersubstitutions of type τ = (n)." Discussiones Mathematicae - General Algebra and Applications 30.2 (2010): 173-191. <http://eudml.org/doc/276468>.

@article{WattapongPuninagool2010,

abstract = {A (usual) hypersubstitution of type τ is a function which takes each operation symbol of the type to a term of the type, of the same arity. The set of all hypersubstitutions of a fixed type τ forms a monoid under composition, and semigroup properties of this monoid have been studied by a number of authors. In particular, idempotent and regular elements, and the Green’s relations, have been studied for type (n) by S.L. Wismath.
A generalized hypersubstitution of type τ=(n) is a mapping σ which takes the n-ary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ=(n), and any two such extensions can be composed in a natural way. Thus, the set $Hyp_\{G\}(n)$ of all generalized hypersubstitutions of type τ=(n) forms a monoid. In this paper we study the semigroup properties of $Hyp_\{G\}(n)$. In particular, we characterize the idempotent and regular generalized hypersubstitutions, and describe some classes under Green’s relations of this monoid.},

author = {Wattapong Puninagool, Sorasak Leeratanavalee},

journal = {Discussiones Mathematicae - General Algebra and Applications},

keywords = {monoid; regular elements; idempotent elements; Green's relations; generalized hypersubstitution},

language = {eng},

number = {2},

pages = {173-191},

title = {The monoid of generalized hypersubstitutions of type τ = (n)},

url = {http://eudml.org/doc/276468},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Wattapong Puninagool

AU - Sorasak Leeratanavalee

TI - The monoid of generalized hypersubstitutions of type τ = (n)

JO - Discussiones Mathematicae - General Algebra and Applications

PY - 2010

VL - 30

IS - 2

SP - 173

EP - 191

AB - A (usual) hypersubstitution of type τ is a function which takes each operation symbol of the type to a term of the type, of the same arity. The set of all hypersubstitutions of a fixed type τ forms a monoid under composition, and semigroup properties of this monoid have been studied by a number of authors. In particular, idempotent and regular elements, and the Green’s relations, have been studied for type (n) by S.L. Wismath.
A generalized hypersubstitution of type τ=(n) is a mapping σ which takes the n-ary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ=(n), and any two such extensions can be composed in a natural way. Thus, the set $Hyp_{G}(n)$ of all generalized hypersubstitutions of type τ=(n) forms a monoid. In this paper we study the semigroup properties of $Hyp_{G}(n)$. In particular, we characterize the idempotent and regular generalized hypersubstitutions, and describe some classes under Green’s relations of this monoid.

LA - eng

KW - monoid; regular elements; idempotent elements; Green's relations; generalized hypersubstitution

UR - http://eudml.org/doc/276468

ER -

## References

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- [2] S. Leeratanavalee and K. Denecke, Generalized Hypersubstitutions and Strongly Solid Varieties, p. 135-145 in: General Algebra and Applications, Proc. of the '59 th Workshop on General Algebra', '15 th Conference for Young Algebraists Potsdam 2000', Shaker Verlag 2000.
- [3] S. Leeratanavalee, Submonoids of Generalized Hypersubstitutions, Demonstratio Mathematica XL (1) (2007), 13-22. Zbl1120.08001
- [6] W. Puninagool and S. Leeratanavalee, The Order of Generalized Hypersubstitutions of Type τ =(2), International Journal of Mathematics and Mathematical Sciences, Vol 2008 (2008), Article ID 263541, 8 pages. doi: 10.1155/2008/263541 Zbl1159.08002
- [7] W. Taylor, Hyperidentities and Hypervarieties, Aequationes Mathematicae 23 (1981), 111-127. Zbl0491.08009
- [8] S.L. Wismath, The monoid of hypersubstitutions of type (n), Southeast Asian Bull. Math. 24 (1) (2000), 115-128. Zbl0961.08007

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