Document Type : Research Paper


Department of Mathematics, Faculty of Science, University of Farhangian , Tabriz, Iran.


The main result of this paper is a characterization of the strongly algebraically closed algebras in the  lattice of all real-valued continuous functions and the equivalence classes of $\lambda$-measurable. We shall provide conditions  which strongly algebraically closed algebras carry a strictly positive Maharam submeasure. Particularly, it is proved that if $B$ is a strongly algebraically closed lattice  and $(B,\, \sigma)$ is a Hausdorff space  and $B$ satisfies  the   $G_\sigma$ property, then $B$ carries a strictly positive Maharam submeasure.


[1] B. Balcar, T. Jech and T. Pazak, Complete ccc Booleab algebras, the order sequential topology, and a problem of Von Neumann, Bull. London Math. Soc., 37 (2005), pp. 885-898.
[2] E. Behrends, $L^p$-Struktur in Banachraumen, Studia Math., 55 (1976), pp. 71-85.
[3] G. Birkhoff, Lattice theory, Colloq. Publ., Vol. 25, Amer. Math. Soc, Providence, R. I., 1967.
[4] F. Cunningham, $L$-structure in $L$-spaces, Trans. Amer. Math. Soc., 95 (1960), pp. 274-299.
[5] E. Daniyarova, A. Miasnikov, and V. Remeslennikov, Unification theorems in algebraic geometry, Algebra and Discrete Mathematics, 1 (2008), pp. 80-112.
[6] S. Givant and P. Halmos, Introduction to Boolean algebras, Springer Science$+$ Business Media, New York, 2009.
[7] V.A. Gorbunov, Algebraic theory of quasivarieties, Nauchnaya Kniga, Novosibirsk, 1999; English transl., Plenum, 1998.
[8] G. Gratzer, Universal algebra, Van Nostrand, Princeton, N. J., 2008.
[9] G. Higman and E.L. Scott, Existentially Closed Groups, Clarendon Press, 1988.
[10] W. Hodges, Model theory, University Press, Cambridge, 1993.
[11] A.G. Kusraev and S.S. Kutateladze, Nonstandard Methods of Analysis [in Russian], Nauka, Novosibirsk (1990).
[12] D. Maharam, An algebraic characterization of measure algebras, Ann. of Math., 48 (1947), pp. 154-167.
[13] A. Molkhasi, On strrongly algebraically closed lattices, J. Sib. Fed. Univ. Math. Phys., 9 (2016), pp. 202-208.
[14] A. Myasnikov and V. Remeslennikov, Algebraic geometry over groups II: logical foundations, J. Algebra, 234 (2000), pp. 225-276.
[15] B. Plotkin, Algebras with the same (algebraic) geometry, Proc. Steklov Inst. Math., 242 (2003), pp. 165-196.
[16] J. Schmid, Algebraically and existentially closed distributive lattices, Zeilschr f. miath. Logik und Crztndlagen d. Math. Bd., 25 (1979), pp. 525-530.
[17] W.R. Scott, Algebraically closed groups, Proc. Amer. Math. Soc., 2 (1951), pp. 118-121.
[18] A. Shevlyakov, Algebraic geometry over Boolean algebras in the language with constants, J. Math. Sciences, 206 (2015), pp. 724-757.
[19] R. Sikorski, Boolean Algebras, Springer-Verlag, Berlin etc., 1964.
[20] D.A. Vladimirov, Boolean algebras, Nauka, Moscow, 1969.