Free Modal Pseudocomplemented De Morgan Algebras

Abstract

<jats:p>Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].</jats:p>

Symmetric operators on modal pseudocomplemented De Morgan algebras

“…In this subsection, we will show that Theorem 5.1 has particular cases that was previously obtained in the literature in [19,15].…”

“…In what follows, we will study the prime spectrum of a given C k -algebra A with an algebraic technique. First, let us observe the study could be done using topological representation through the techniques given in [16,18]. Proposition 2.5 Let (A, t) be a C k -algebra and U ⊆ A an ultrafilter.…”

“…Later, he showed that every mpM −algebra is a TMA by defining ∇x = ∼ (∼ x∧x * ) and △x =∼ ∇ ∼ x. In [18] (see also [19]), the authors have proven that the subdirectly irreducible mpM −algebras are three as TMAs, in fact, Hasse diagrams are the same in each case, but 3-chain-mpM −algebra is not a subalgebra of four-elements. The mentioned algebras are the following: T 2 = {0, 1} with 0 < 1, ∼ 0 = 0 * = 1, ∼ 1 = 1 * = 0; T 3 = {0, a, 1}, with 0 < a < 1, ∼ a = a, a * = 0, ∼ 0 = 0 * = 1, ∼ 1 = 1 * = 0; T 4 = {0, a, b, 1} with a ≤ b, b ≤ a and 0 < a, b < 1, ∼ b = a * = b, ∼ a = b * = a, ∼ 0 = 0 * = 1, ∼ 1 = 1 * = 0.…”

“…Sometimes we write "tx" instead of "t(x)". It is worth mentioning that the class C k -algebra is a variety; and, as examples, we have the classes 1-cyclic mpM -algebra and 2-cyclic mpM -algebra were studied in [18] and [15], respectively. Besides, if an mpMalgebra is a 3-valued Lukasiewicz algebra then the k-cyclic version is a special class studied in [32,33,12].…”

A note on k-cyclic modal pseudocomplemented De Morgan algebras

“…In [13], it was showed that every mpM −algebra is a TMA-algebra by defining ∇x = ∼ (∼ x ∧ x * ) and △x =∼ ∇ ∼ x, but the varieties are not equivalent as it was shown in [14,Section 5]. In [14] (see also [15]), the authors proved that the subdirectly irreducible mpM −algebras are three as displayed in the following Remark.…”