# Pseudocomplements in sum-ordered partial semirings

Discussiones Mathematicae - General Algebra and Applications (2007)

- Volume: 27, Issue: 2, page 169-186
- ISSN: 1509-9415

## Access Full Article

top## Abstract

top## How to cite

topJānis Cīrulis. "Pseudocomplements in sum-ordered partial semirings." Discussiones Mathematicae - General Algebra and Applications 27.2 (2007): 169-186. <http://eudml.org/doc/276896>.

@article{JānisCīrulis2007,

abstract = {We study a particular way of introducing pseudocomplementation in ordered semigroups with zero, and characterise the class of those pseudocomplemented semigroups, termed g-semigroups here, that admit a Glivenko type theorem (the pseudocomplements form a Boolean algebra). Some further results are obtained for g-semirings - those sum-ordered partially additive semirings whose multiplicative part is a g-semigroup. In particular, we introduce the notion of a partial Stone semiring and show that several well-known elementary characteristics of Stone algebras have analogues for such semirings.},

author = {Jānis Cīrulis},

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

keywords = {Glivenko theorem; partial monoid; partial semiring; pseudocomplementation; semigroup; Stone semiring; sum-ordering},

language = {eng},

number = {2},

pages = {169-186},

title = {Pseudocomplements in sum-ordered partial semirings},

url = {http://eudml.org/doc/276896},

volume = {27},

year = {2007},

}

TY - JOUR

AU - Jānis Cīrulis

TI - Pseudocomplements in sum-ordered partial semirings

JO - Discussiones Mathematicae - General Algebra and Applications

PY - 2007

VL - 27

IS - 2

SP - 169

EP - 186

AB - We study a particular way of introducing pseudocomplementation in ordered semigroups with zero, and characterise the class of those pseudocomplemented semigroups, termed g-semigroups here, that admit a Glivenko type theorem (the pseudocomplements form a Boolean algebra). Some further results are obtained for g-semirings - those sum-ordered partially additive semirings whose multiplicative part is a g-semigroup. In particular, we introduce the notion of a partial Stone semiring and show that several well-known elementary characteristics of Stone algebras have analogues for such semirings.

LA - eng

KW - Glivenko theorem; partial monoid; partial semiring; pseudocomplementation; semigroup; Stone semiring; sum-ordering

UR - http://eudml.org/doc/276896

ER -

## References

top- [1] G. Birkhoff, Lattice Theory, AMS Colloq. Publ. 25, Providence, Rhode Island 1967.
- [2] T.S. Blyth, Pseudo-residuals in semigroups, J. London Math. Soc. 40 (1965), 441-454. Zbl0136.26903
- [3] J. Cīrulis, Quantifiers in semiring like logics, Proc. Latvian Acad. Sci. 57 B (2003), 87-92. Zbl1043.03047
- [4] A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures, Kluwer Acad. Publ. and Ister Science, Dordrecht, Bratislava e.a., 2000. Zbl0987.81005
- [5] O. Frink, Representation of Boolean algebras, Bull. Amer. Math. Soc. 47 (1941), 755-756. Zbl0063.01459
- [6] O. Frink, Pseudo-complements in semilattices, Duke Math. J. 29 (1962), 505-514. Zbl0114.01602
- [7] G. Grätzer, General Lattice Theory, Akademie-Verlag, Berlin, 1978. Zbl0436.06001
- [8] U. Hebisch and H.J. Weinert, Semirings. Algebraic Theory and Applications in Computer Science, World Scientific, Singapore e.a., 1993. Zbl0829.16035
- [9] J.M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford 1995.
- [10] M. Jackson and T. Stokes, Semilattice pseudo-complements on semigroups, Commun. Algebra 32 (2004), 2895-2918. Zbl1069.20055
- [11] M.F. Janowitz and C.S. Johnson, Jr., A note on Brouwerian and Glivenko semigroups, J. London Math. Soc. (2) 1 (1969), 733-736. Zbl0185.04803
- [12] E.G. Manes and D.B. Benson, The inverse semigroup of a sum-ordered semiring, Semigroup Forum 31 (1985), 129-152. Zbl0549.06014
- [13] T.P. Speed, A note on commutative semigroups, J. Austral. Math. Soc. 111 (1968), 731-736. Zbl0172.02502
- [14] U.M. Swamy and G.C. Rao e.a., Birkhoff centre of a poset, South Asia Bull. Math. 26 (2002), 509-516. Zbl1017.06002

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.