The Sheffer stroke operation reducts of basic algebras

Abstract In this study, a term operation Sheffer stroke is presented in a given basic algebra 𝒜 and the properties of the Sheffer stroke reduct of 𝒜 are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.


Introduction
Take into consideration the problem of expressing equational concepts as simply as possible with the least number of operations or the least number of axioms and so forth. For an example about the least number of axioms, Tarski solved the related problem for Abelian groups with the single axiom x=.y=.z=.x=y/// D z in terms of the division operation in 1938 [1]. As a typical example, we tackle the use of Sheffer stroke operation in algebraic structures. The Sheffer stroke term operation was firstly given by H. M. Sheffer in 1913 [2]. He proved that all Boolean functions could be transplanted to a single binary operation for term operations. In recent years, the problem for Boolean algebra was solved with a single axiom in terms of the Sheffer Stroke operation [3].
The reduction attempts interest mathematicians who want to use less operations or axioms or formulas for the structures under their considerations. The first implication reduct of Boolean algebras connectives was studied by J. C. Abbott [4] under the name implication algebra. Once the logic of quantum mechanics was axiomatized by means of orthomodular lattices, Abbott obtained implication reducts of orthomodular lattices, called orthoimplication algebras in [5]. Later on, this work was generalized for implication reducts of ortholattices by Chajda and Halaš [6], and by Chajda [7] to orthomodular lattices but without the compatibility condition in Chajda, Halaš and Länger [8].
Basic algebras were introduced in Chajda and Emanovský [9], see also Chajda [10] and Chajda et al. [11] and [12,13] for further information. Basic algebras are an important concept used in different non-classical logics since they contain orthomodular lattices L D .LI _;^; ? ; 0; 1/, where x˚y D .x^y ? /_y and :x D x ? , and constitute as well as provide an axiomatization of the logic of quantum mechanics along with MV-algebras [14], which get an axiomatization of many-valued Łukasiewicz logics; see Chajda [15] and Chajda et al. [16].
Given that the connective Sheffer stroke operation plays a central role in all mentioned logics above, in general, we would like to characterize this operation in basic algebras.

Preliminaries
The following fundamental notions are taken from [17] and [18]. Definition 2.1. A bounded lattice is an algebraic structure L D .LI _;^; 0; 1/ such that L D .LI _;^/ is a lattice having the following properties: (i) for all x 2 L, x _ 1 D 1 and x^1 D x, (ii) for all x 2 L, x _ 1 D x and x^0 D 0. The elements 0 and 1 are called the least element and the greatest element of the lattice, respectively. Definition 2.2. Let L D .LI _;^/ be a lattice. A mapping x 7 ! x ? is called an antitone involution if it satisfies the following: D .x _ y/ ? : The operation j is said to be a Sheffer stroke operation if it satisfies the following conditions: .S1/ xjy D yjx; .S2/ .xjx/j.xjy/ D x; .S3/ xj..yjz/j.yjz// D ..xjy/j.xjy//jz; .S4/ .xj..xjx/j.yjy///j.xj..xjx/j.yjy/// D x: If additionally it satisfies the identity .S5/ yj.xj.xjx// D yjy; it is called an ortho Sheffer stroke operation.  In order to obtain a construction of Sheffer stroke reduction of basic algebras, we firstly give the definition of a basic algebra: As shown in [20], every basic algebra can be thought alternatively as a bounded lattice with section antitone involutions. Now, we are dealing with in the case when for an element x of a basic algebra A the negation :x is a complement of x in the induced lattice L.A/. Then we can give the following lemma and its corollary:

The Sheffer Stroke Reduction of Basic Algebras
As mentioned in [21], we can consider A D .AI˚; :; 0/ alternatively in signature f!; 0g, where x ! y D :x˚y. When regarding a logic axiomatized by a basic algebra, especially if A is an MV-algebra, then it is a many-valued fuzzy logic, or if A is an orthomodular lattice, it is a logic of quantum mechanics. Then the operation ! plays the role of logic connective implication. There are also some interesting results related to implication operation used (see [22][23][24][25][26][27]). Starting from this point of view, we construct an alternative signature fjg which consists of only the Sheffer stroke operation. Hence, it is of some importance to investigate the Sheffer stroke reduct in a general setting for basic algebras. Now, we can define the following concept. First of all, we give some simple properties of Sheffer Stroke basic algebras. Then, we demonstrate that every such algebra has an algebraic constant 1 as is the case for implication basic algebras [20].
.v/ W By using .i/ in .SH 3/, we get ...xj.yjy//j.yjy//j.zjz//j.xj.zjz// D 1: Proof. To prove this claim, we construct a model for each axiom in which that axiom is false while the others are true. Let A D .f0; 1g; j A / be our model defined in the following tables: .1/Independence of .SH1/: We define the operation j A in the following Then j A satisfies .SH1/ and .SH 3/, but not .SH 2/ because if we choose x D 1 and y D 0, then .1j.0j0//j.0j0/ D 1 ¤ 0 D .0j.1j1//j.1j1/.
.3/Independence of .SH 3/: Define the operation j A as in Table 3: The model j A satisfies .SH1/ and .SH 2/, but not .SH 3/. When we choose x D 0 and y D z D 1, we get ...0j.1j1//j.1j1//j.1j1//j..0j.1j1//j.0j.1j1/// D 1 ¤ 0 D 0j.0j0/: To construct a bridge between basic algebras and Sheffer stroke basic algebras, we need the following theorem: To reveal the structure of Sheffer stroke basic algebras, we introduce a partial order relation on A.
Lemma 3.5. Let .AI j/ be a Sheffer stroke basic algebra. A binary relation Ä is defined on A as follows: x Ä y if and only if xj.yjy/ D 1: Then the binary relation Ä is a partial order on A such that x Ä 1 for each x 2 A. Moreover, we have z Ä .xj.zjz// and x Ä y implies yj.zjz/ Ä xj.zjz/ for all x; y; z 2 A. Proof.
Theorem 3.6. Let .AI j/ be a Sheffer stroke basic algebra and Ä its induced partial order. Then .AI Ä/ is a join semi-lattice with the greatest element 1 where x _ y D .xj.yjy//j.yjy/.
Hence .xj.yjy//j.yjy/ is an upper bound for x and y.
Let .AI j/ be a Sheffer stroke basic algebra. The semilattice .AI _/ derived in the above theorem will be called the induced semilattice of .AI j/. This is the infimum of x; y 2 OEp; 1. Consequently, .OEp; 1I _;^p; p / is a lattice with an antitone involution.
Corollary 3.8. Let .AI j/ be a Sheffer Stroke basic algebra and Ä is the induced partial order on this system. Then .AI Ä/ is a join-semilattice which has the greatest element 1. For each p 2 A the closed interval OEp; 1 is a basic algebra .OEp; 1I˚p; : p ; p/ if x˚p y D .xj.pjp//j.yjy/ and : p x D xj.pjp/ are defined for all x; y 2 A.
From now on, .OEp; 1I˚p; : p ; p/ is said to be an interval basic algebra with the greatest element 1 and the least element p. Therefore, Theorem 3.7 corresponds to the semilattice structure of a Sheffer stroke basic algebra. We prove that this explanation is complete, in other words, the other direction of Theorem 3.7 can be obtained.
Theorem 3.9. Let .AI _; 1/ be a join-semilattice with the greatest element 1 such that for every p 2 A, the closed interval OEp; 1 is a lattice with antitone involution x 7 ! x p . If we define xjy D .x _ y p / y p , then .AI j/ is a Sheffer Stroke basic algebra.
We say that .AI j/ is a Sheffer Stroke basic algebra which has the least element if there exists an element 0 2 A such that 0 Ä a for all a 2 A, where Ä is the induced partial order. So, any Sheffer Stroke basic algebra with the least element 0 satisfies the identity 0j.xjx/ D 1.
The proof of the following theorem is straightforward.
Theorem 3.10. Let .AI j/ be a Sheffer Stroke basic algebra which has the least element 0. If we define :x D xj.0j0/ and x˚y D .xj.0j0//j.yjy/, then the system .AI˚; :; 0/ is a basic algebra and xjy D :x˚:y.
In the remaining part of this work, we show that there is a bridge between Sheffer stroke basic algebras and Boolean algebras. .(W/ It is verified similarly.
Theorem 3.14. Let .AI j/ be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and .AI _;^0; 0 ; 0; 1/ its induced complemented lattice with an antitone involution x 7 ! x 0 . Then, there exists unique x 0 such that x _ x 0 D 1 and x^0 x 0 D 0 for all x 2 A.
Proof. We show that there exists x 7 ! x 0 an antitone involution of A such that x 0 _ x D 1 and x 0^0 x D 0 in Lemma 3.11. For the uniqueness, assume that x 0 D k and x 0 D l. Then by Lemma 3.13 .vi /, we have k D x 0 D .xjx/ and l D x 0 D .xjx/: Then from these equalities we get x _ k D 1 ) .xjx/j.kjk/ D 1 ) lj.kjk/ D 1 ) l Ä k: Using the same technique, we can obtain k Ä l. Therefore, k D l. Hence, we have unique x 0 such that x 0 _ x D 1 for each x 2 A. The identity x^0 x 0 D 0 is verified similarly. Lemma 3.16. Let .AI j/ be a commutative basic algebra. Then the interval basic algebra .OEp; 1I˚p; : p ; p/ is commutative for each p 2 A.
From Theorem 2:8 in [20] we obtain the following corollary: Corollary 3.17. Let .AI j/ be a commutative Sheffer Stroke basic algebra and .AI _/ its induced semilattice. Then (i) the interval basic algebra .OEp; 1I˚p; : p ; p/ is commutative basic algebra for each p 2 A, (ii) the interval lattice .OEp; 1; _;^p/ is distributive for each p 2 A.
Proof. It follows from Theorem 3.10 and Corollary 2.10.