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Filters in Residuated Skew Lattices: Part Ii Honam Mathematical J. 40 (2018), No. 3, pp. 401–431 https://doi.org/10.5831/HMJ.2018.40.3.401 (SKEW) FILTERS IN RESIDUATED SKEW LATTICES: PART II R. Koohnavard and A. Borumand Saeid∗ Abstract. In this paper, some kinds of (skew) filters are defined and are studied in residuated skew lattices. Some relations are got between these filters and quotient algebras constructed via these filters. The Green filter is defined which establishes a connection between residuated lattices and residuated skew lattices. It is in- vestigated that relationships between Green filter and other types of filters in residuated skew lattices and the relationship between residuated skew lattice and other skew structures are studied. It is proved that for a residuated skew lattice, skew Hilbert algebra and skew G-algebra are equivalent too. 1. Introduction Skew lattices were introduced for the first time by Jordan [10]. The study of skew lattices began with the 1989 paper of Leech [16]. Skew lattices are a generalization of lattices that operations _; ^ are not com- mutative. In skew lattices, two different order concepts can be defined: the natural preorder, denoted by and the natural partial order denoted by ≤. For given a skew lattice (A; _; ^) both algebraic reducts (A; _) and (A; ^) are bands, that is semigroups whose elements are idempotent (x2 = x). J. Leech explained skew Boolean algebras are non-commutative one- pointed generalisations of (possibly generalised) Boolean algebras [14], each right-handed skew Boolean algebra can be embedded into a generic skew Boolean algebra of partial functions from a given set to the codomain Received October 28, 2017. Revised May 12, 2018. Accepted May 21, 2018. 2010 Mathematics Subject Classification. 03G10, 06A75, 03G25. Key words and phrases. residuated skew lattice, (Stonean, (positive) implicative, fantastic, easy) (skew) filter. *Corresponding author 402 R. Koohnavard and A. Borumand Saeid f0; 1g [7], also defined normal and conormal skew lattices [12]. K. Cvetko- Vah defined skew Heyting algebra as dual skew Boolean algebra [7]. J. Pita Costa, expressed the notion of (skew) ideals and (skew) filters in skew lattices [18]. A. Borumand Saeid and R. Koohnavard defined residuated skew lattices as non-commutative generalization of residu- ated lattices. They defined deductive system and skew deductive system in residuated skew lattices and showed that the class of all conormal residuated skew lattices that x ! y = (y _ x _ y) !y y forms a variety too. They showed that Green’s relation D is a congruence relation on residuated skew lattice and its quotient algebra is a residuated lattice. The authors investigated that for any filter F of a residuated skew lat- tice A can associate a congruence and conversely [2]. We showed that in residuated skew lattice, deductive system and filter are equivalent, but skew deductive system and skew filter are not equivalent and any skew deductive system is a skew filter under some conditions [11]. For better underestanding of residuated skew lattices, we investigate filters and use them for classification and obtain residuated skew lattices structural properties. The filter theory in residuated skew lattices plays an important role. From a logic point of view, various filters have natural interpretation as various sets of provable formulas. In this direction, we define some types of (skew) filters in residuated skew lattice and investigate relationships among them. Also we define Green filter and show that quotient algebra via this filter is a residuated lattice and we state and prove the relationship between Green filter and other types of filters. We get some relations between other types of filters and their quotient algebras, for example if F is an easy filter, then A=F is a skew semi-G-algebra. We prove that A is a skew G-algebra iff any filter of A is an implicative filter and for a residuated skew lattice, skew Hilbert algebra and skew G-algebra are equivalent. We show that skew Boolean algebra is not a residuated skew lattice. 2. Preliminaries In this section, we review some properties of skew lattices and resid- uated skew lattices which we need in the sequel. Definition 2.1 ([1]). A skew lattice is an algebra (A; _; ^) of type (2; 2) such that satisfies in the following identities: (i)( x _ y) _ z = x _ (y _ z) and (x ^ y) ^ z = x ^ (y ^ z), (ii) x ^ x = x and x _ x = x, (Skew) filters in residuated skew lattices: Part II 403 (iii) x ^ (x _ y) = x = x _ (x ^ y) and (x ^ y) _ y = y = (x _ y) ^ y. The identities found in (i − iii) are known as the associative law, the idempotent laws and absorption laws respectively. In view of the associativity (i), we can omit parentheses when no ambiguity arises. On a given skew lattice A the natural partial order ≤ and natural preorder respectively are defined by x ≤ y iff x ^ y = x = y ^ x or dually x _ y = y = y _ x and x y if and only if y _ x _ y = y or equivalently x ^ y ^ x = x. Relation D is defined by x D y iff x _ y _ x = x and y _ x _ y = y or dually, x ^ y ^ x = x and y ^ x ^ y = y. D is called the natural equivalence and it coincides with Green’s relation D on both semigroups (A; ^) and (A; _) [5]. For any elements x; y of a skew lattice A, x D y iff x _ y = y ^ x [5]. A pair of natural congruences, L and R, refine D . We say that x is L- related to y (denoted xLy) if x^y = x and y^x = y, or dually, x_y = y and y _ x = x. Likewise, x and y are R-related (x R y) if x ^ y = y and y^x = x, or dually, x_y = x and y_x = y. A skew lattice is left-handed if D = L so that x ^ y = x = y _ x on each rectangular subalgebra and is right-handed if D = R so that x ^ y = y = y _ x on each rectangular subalgebra [5]. Left-handed skew lattices can be defined by similar identities: x _ y _ x = y _x or equivalently x^y ^x = x^y and right-handed skew lattices are the ones satisfying the identity x ^ y ^ x = y ^ x or equivalently x _ y _ x = x _ y [18]. Definition 2.2 ([12]). Skew lattice A is normal if x ^ y ^ z ^ w = x ^ z ^ y ^ w and is conormal if x _ y _ z _ w = x _ z _ y _ w for all x; y; z; w 2 A. Proposition 2.3 ([12]). A skew lattice A is normal iff each sub skew lattice (# x) is a sub lattice of A. Dually, A is conormal iff each sub skew lattice (" x) is a sub lattice of A ((" x) = fy 2 Ajy ≥ xg; (# x) = fy 2 Ajy ≤ xg). A skew lattice is distributive if it satisfies x ^ (y _ z) ^ x = (x ^ y ^ x) _ (x ^ z ^ x) and x _ (y ^ z) _ x = (x _ y _ x) ^ (x _ z _ x) [13]. A skew lattice is called co-strongly distributive if it satisfies the identities: x _ (y ^ z) = (x _ y) ^ (x _ z) and (x ^ y) _ z = (x _ z) ^ (y _ z). A skew lattice is called strongly distributive if it satisfies the identities: x ^ (y _ z) = (x ^ y) _ (x ^ z) and (x _ y) ^ z = (x ^ z) _ (y ^ z). Any co-strongly distributive skew lattice or strongly distributive skew lattice is a distributive skew lattice [7]. 404 R. Koohnavard and A. Borumand Saeid Proposition 2.4 ([7]). A skew lattice A is co-strongly distributive if and only if A is jointly : (i) quasi-distributive: the maximal lattice image A=D is a distributive lattice, (ii) symmetric: x ^ y = y ^ x if and only if x _ y = y _ x, (iii) conormal: x _ y _ z _ w = x _ z _ y _ w. All distributive skew lattices are quasi-distributive [13]. A skew chain is a skew lattice that x y or y x, for all x; y 2 A. If A is a skew chain, then A=D is a chain [7]. Theorem 2.5 ([15]). A skew lattice A is strongly distributive iff it is distributive, normal and symmetric. Definition 2.6 ([15]). A skew Boolean algebra is an algebra (A; _; ^; n; 0) such that (i)( A; _; ^; 0) is a strongly distributive skew lattice with 0, (ii) n is a binary operation satisfying (x ^ y ^ x) _ (xny) = x and (x ^ y ^ x) ^ (xny) = 0. Definition 2.7 ([7]). A skew Heyting algebra is an algebra A = (A; _; ^; !; 1) of type (2; 2; 2; 0) satisfying the following: (i)( A; _; ^; 1) is a co-strongly distributive skew lattice with top 1. Thus any (" u) is a bounded distributive lattice, (ii) For any u 2 A, an operation !u can be defined on (" u) such that (" u; _; ^; !u; u; 1) is a Heyting algebra with top 1 and bottem u, (iii) x ! y = y _ x _ y !y y. A primitive skew lattice is a skew lattice that has exactly two com- parable D-class A B. A band is a semigroup that its elements are idempotent. A rectangular band is a band that xyx = x or equivalently xyz = xz. A skew lattice is rectangular, if (A; _) or (A; ^) is rectangular band [15].
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