Previous |  Up |  Next


bounded lattice; lattice ordered group; generalized cardinal property; homogeneity
We denote by $K$ the class of all cardinals; put $K^{\prime }= K \cup \lbrace \infty \rbrace $. Let $\mathcal C$ be a class of algebraic systems. A generalized cardinal property $f$ on $\mathcal C$ is defined to be a rule which assings to each $A \in \mathcal C$ an element $f A$ of $K^{\prime }$ such that, whenever $A_1, A_2 \in \mathcal C$ and $A_1 \simeq A_2$, then $f A_1 =f A_2$. In this paper we are interested mainly in the cases when (i) $\mathcal C$ is the class of all bounded lattices $B$ having more than one element, or (ii) $\mathcal C$ is a class of lattice ordered groups.
[1] G.  Birkhoff: Lattice Theory. Third Edition, Providence, 1967. MR 0227053 | Zbl 0153.02501
[2] P.  Conrad: Lattice Ordered Groups. Tulane University, 1970. Zbl 0258.06011
[3] E. K.  van Douwen: Cardinal functions on compact $F$-spaces and weakly complete Boolean algebras. Fundamenta Math. 113 (1982), 235–256.
[4] E. K.  van Douwen: Cardinal functions on Boolean spaces. In: Handbook of Boolean Algebras, J. D.  Monk and R.  Bonnet (eds.), North Holland, Amsterdam, 1989, pp. 417–467. MR 0991599
[5] J.  Jakubík: Konvexe Ketten in $\ell $-Gruppen. Časopis pěst. mat. 84 (1959), 53–63. MR 0104740
[6] J.  Jakubík: Cardinal properties of lattice ordered groups. Fundamenta Math. 74 (1972), 85–98. DOI 10.4064/fm-74-2-85-98 | MR 0302528
[7] J.  Jakubík: Radical classes of generalized Boolean algebras. Czechoslovak Math.  J. 48 (1998), 253–268. DOI 10.1023/A:1022885303504 | MR 1624315
[8] J. D.  Monk: Cardinal functions on Boolean algebras. In: Orders, Description and Roles, M. Pouzet and D. Richard (eds.), North Holland, Amsterdam, 1984, pp. 9–37. MR 0779843 | Zbl 0557.06009
[9] R. S.  Pierce: Some questions about complete Boolean algebras. Proc. Symp. Pure Math., Vol.  II, Lattice Theory, Amer. Math. Soc., Providence, 1961. MR 0138570 | Zbl 0101.27104
[10] F.  Šik: Über subdirekte Summen geordneter Gruppen. Czechoslovak Math.  J. 10 (1960), 400–424. MR 0123626
Partner of
EuDML logo