Geodesic
Pseudocomplements in sum-ordered partial semirings
Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 2, pp. 169-186.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: Glivenko theorem, partial monoid, partial semiring, pseudocomplementation, semigroup, Stone semiring, sum-ordering 
@article{DMGAA_2007_27_2_a1,
     author = {C\={i}rulis, J\={a}nis},
     title = {Pseudocomplements in sum-ordered partial semirings},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {169--186},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a1/}
}
TY  - JOUR
AU  - Cīrulis, Jānis
TI  - Pseudocomplements in sum-ordered partial semirings
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2007
SP  - 169
EP  - 186
VL  - 27
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a1/
LA  - en
ID  - DMGAA_2007_27_2_a1
ER  - 
%0 Journal Article
%A Cīrulis, Jānis
%T Pseudocomplements in sum-ordered partial semirings
%J Discussiones Mathematicae. General Algebra and Applications
%D 2007
%P 169-186
%V 27
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a1/
%G en
%F DMGAA_2007_27_2_a1
Cīrulis, Jānis. Pseudocomplements in sum-ordered partial semirings. Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 2, pp. 169-186. http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a1/

[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.

[3] J. Cīrulis, Quantifiers in semiring like logics, Proc. Latvian Acad. Sci. 57 B (2003), 87-92.

[4] A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures, Kluwer Acad. Publ. and Ister Science, Dordrecht, Bratislava e.a., 2000.

[5] O. Frink, Representation of Boolean algebras, Bull. Amer. Math. Soc. 47 (1941), 755-756.

[6] O. Frink, Pseudo-complements in semilattices, Duke Math. J. 29 (1962), 505-514.

[7] G. Grätzer, General Lattice Theory, Akademie-Verlag, Berlin, 1978.

[8] U. Hebisch and H.J. Weinert, Semirings. Algebraic Theory and Applications in Computer Science, World Scientific, Singapore e.a., 1993.

[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.

[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.

[12] E.G. Manes and D.B. Benson, The inverse semigroup of a sum-ordered semiring, Semigroup Forum 31 (1985), 129-152.

[13] T.P. Speed, A note on commutative semigroups, J. Austral. Math. Soc. 111 (1968), 731-736.

[14] U.M. Swamy and G.C. Rao e.a., Birkhoff centre of a poset, South Asia Bull. Math. 26 (2002), 509-516.