A classification of hull operators in archimedean lattice-ordered groups with unit

Document Type : Research Paper


1 Department of Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA.

2 Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.



The category, or class of algebras, in the title is denoted by $\bf W$. A hull operator (ho) in $\bf W$ is a reflection in the category consisting of $\bf W$ objects with only essential embeddings as morphisms. The proper class of all of these is $\bf hoW$. The bounded monocoreflection in $\bf W$ is denoted $B$. We classify the ho's by their interaction with $B$ as follows. A ``word'' is a function $w: {\bf hoW} \longrightarrow {\bf W}^{\bf W}$ obtained as a finite composition of $B$ and $x$ a variable ranging in $\bf hoW$. The set of these,``Word'', is in a natural way a partially ordered semigroup of size $6$, order isomorphic to ${\rm F}(2)$, the free $0-1$ distributive lattice on $2$ generators. Then, $\bf hoW$ is partitioned into $6$ disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic order-preserving quotient of Word ($\approx {\rm F}(2)$). Of the $6$: $1$ is of size $\geq 2$, $1$ is at least infinite, $2$ are each proper classes, and of these $4$, all quotients are chains; another $1$ is a proper class with unknown quotients; the remaining $1$ is not known to be nonempty and its quotients would not be chains.


[1] Anderson, M.E. and Feil, T.H., "Lattice-ordered Groups: an Introduction", Springer Science & Business Media, 2012.
[2] Balbes, R. and Dwinger, P., "Distributive lattices", New York, 1974.
[3] Ball, R. and Hager, A.W., Algebraic extensions of an archimedean lattice-ordered group I, J. Pure Appl. Algebra 85(1) (1993), 1-20.
[4] Ball, R., Hager, A.W., and Neville, C., The Quasi-FK cover of Compact Hausdorff Space and the AC-Ideal Completion of an Archimedean l-Group, General Top. Appl.: Proceedings of the 1988 Northeast Conference 123, 1990.
[5] Banaschewski, B. and Hager, A., Representation of H-closed monoreflections in archimedean l-groups with weak unit, Categ. General Alg. Struct. Appl. 9(1) (2018), 1-13.
[6] Bigard, A., Keimel, K., and Wolfenstein, S., "Groupes et Anneaux R´eticul´es", Springer, 2006.
[7] Birkhoff, G., "Lattice theory", American Mathematical Society, 1940.
[8] Carrera, R., Various Completeness of an Archimedean Lattice-ordered Group, 2014.
[9] Carrera, R. and Hager, A.W., On hull classes of l-groups and covering classes of spaces, Math. Slovaca 61(3) (2011), 411-428.
[10] Carrera, R.E. and Hager, A.W., B-saturated hull classes in `-groups and covering classes of spaces, Appl. Categ. Structures, 23(5) (2015), 709-723.
[11] Carrera, R.E. and Hager, A.W., Bounded equivalence of hull classes in archimedean lattice-ordered groups with unit, Appl. Categ. Structures 24(2) (2016), 163-179.
[12] Conrad, P., The essential closure of an archimedean lattice-ordered group, Duke Math. J. 38(1) (1971), 151-160.
[13] Conrad, P., The hulls of representable l-groups and f-rings, J. Austral. Math. Soc. 16(4) (1973), 385-415.
[14] Darnel M. R., "The Theory of Lattice-Ordered Groups", Pure Appl. Math. 187, Marcel Dekker, 1995.
[15] Engelking, R."Outline of General Topology", North-Holland Pub. Co., 1968.
[16] Fine, N., Gillman, L., and Lambek, J., Rings of Quotients of Rings of Functions, McGill University Press, 1996.
[17] Fuchs, L., "Partially Ordered Algebraic Systems", Courier Corporation, 2011.
[18] Gillman, L. and Jerison, M., "Rings of Continuous Functions", Courier Dover Publications, 2017.
[19] Gleason, A., Projective topological spaces, Illinois J. Math. 2(4A) (1958), 482-489.
[20] Hager, A.W., Algebraic closures of l-groups of continuous functions, in "Rings of Continuous Functions", C. Aull, Ed., Lecture Notes Pure Appl. Math. 95 (1985), Marcel Dekker, 165-194.
[21] Hager, A.W., Minimal covers of topological spaces, Ann. New York Acad. Sci. 552(1) (1989), 44-59.
[22] Hager, A.W. and Martinez, J.,"$alpha$-projectable and laterally $alpha$-complete archimedean lattice-ordered groups", in S. Bernahu (ed.), Proc. Conf. on Mem. of T. Retta, Temple U., PA/Addis Ababa (1995), Ethiopian J. Sci., 73–84.
[23] Hager, A.W. and Robertson, L., Representing and ringifying a Riesz space, Sympos. Math 21 (1977), 411-431.
[24] Hager, A.W. and Robertson, L., Extremal units in an archimedean Riesz space, Rendiconti del Seminario matematico della Universit`a di Padova 59 (1978), 97-115.
[25] Henriksen, M., Vermeer, J., and Woods, R., Wallman covers of compact spaces, Instytut Matematyczny Polskiej Akademi Nauk (Warszawa), 1989.
[26] Herrlich, H. and Strecker, G., "Category Theory: an Introduction", Heldermann Verlag, 1979.
[27] Martinez, J., Hull classses of archimedean lattice-ordered groups with unit: A survey", in "Ordered Algebraic Structures", Springer, 2002, 89-121.
[28] Martinez, J., Polar functions—II: Completion classes of archimedean f-algebras vs. covers of compact spaces, J. Pure Appl. Algebra 190(1-3) (2004), 225-249.
[29] Porter, R. and Woods G., "Extensions and absolutes of Hausdorff spaces", Springer Science & Business Media, 2012.
[30] Woods, G., Covering Properties and Coreflective Subcategories a, b, Ann. New York Acad. Sci. 552(1) (1989), 173-184.