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In this paper, the notions of annulets and normal filters are introduced in Stone lattices and their properties are studied. A set of equivalent conditions is obtained to characterize normal filters of a Stone lattice. The extensions of the Glivenko-type congruences on a Stone lattice are investigated via annulets and normal filters. A description of the lattice of all extensions of the Glivenko-type congruences on a Stone lattice is given. A one-to-one correspondence between the class of all extensions and the class of all normal filters of a Stone lattice is obtained. Finally, we observe that every 2 extensions of the Glivenko-type congruences are permutable. PRELIMINARIESIn this section, we recall basic definitions and important results. For more details we refer to previous studies. 2,8,9 Definition 2.1. (Birkhoff 2 ) An algebra (L; ∧, ∨) of type (2,2) is said to be a lattice if for every a, b, c ∈ L, the following properties are satisfied:(1) a ∧ a = a, a ∨ a = a (Idempotency),Math Meth Appl Sci. 2018;41:5719-5732.wileyonlinelibrary.com/journal/mma

This paper is devoted to the study of the class of decomposable double MS-algebras. Necessary and sufficient conditions for a decomposable MS-algebra to be a decomposable double MS-algebra are deduced. We construct decomposable double MS-algebras by means of decomposable MS-quadruples and we prove that there exists a one-to-one correspondence between decomposable double MS-algebras and decomposable MS-quadruples. Moreover, a construction of decomposable K2-algebras (Stone algebras) by means of K2-quadruples (Stone quadruples) is given. We conclude by introducing and characterizing isomorphisms of decomposable double MS-algebras in terms of decomposable MS-quadruples.