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1 Archivu Matheaticu Ivan Chajda Matrix representation of hooorphic appings of finite Boolean algebras Archivu Matheaticu, Vol. 8 (1972), No. 3, Persistent URL: Ters of use: Masaryk University, 1972 Institute of Matheatics of the Acadey of Sciences of the Czech Republic provides access to digitized docuents strictly for personal use. Each copy of any part of this docuent ust contain these Ters of use. This paper has been digitized, optiized for electronic delivery and staped with digital signature within the project DML-CZ: The Czech Digital Matheatics Library

2 ARCH. MATH. 3, SCRIPTA FAC SCI. NAT. UJEP BRUNENSIS, VIII: , 1972 MATRIX REPRESENTATION OF HOMOMORPHIC MAPPINGS OF FINITE BOOLEON ALGEBRAS IVAN CHAJDA (Received February 18, 1972) Soeties it is advantageous to investigate soe of the properties of Boolean algebras on odels. A suitable odel for the investigation of hooorphic appings is just the B-odul defined in [1]. For this reason in this paper I do not deal with an abstract Boolean algebra but with its isoorphic representation the B-odul. 1. Definition 1. Let there be given a set M = {0, 1}. Let us call each eleent of the Cartesian power M the -diensional B-vector (or briefly vector) over M and denote it by sybol a = (a t, a 2,..., a ) where a$ e M for i 1, 2,...,.. Eleents a\ are called coordinates of the B-vector. We shall call the set 9Jl w of all -diensional B-vectores the -diensional B-odul over M. The B-vector a = (a x,..., a ) is equal to B-vector b = (bi,..., b ) just when at = bi for all i. By the su of vectors a, b we call vector c = (c\,..., c ) where ct = a% + bi, for the coordinates holding: = = 0, = 1, 0-+1 = 1+0=1. By the product of vectors a, b we shal call vector d = (di,..., d ), where d% = afii, for the coordinates holding 0.0 = 0, 0.1 = = 1.0 = 0, 1.1 = 1. The vector j = (1, 1,..., 1) is called a unit vector, vector o (0, 0,..., 0) a zero vector. Vector a is called copleent of vector a, if it holds a + a = j, a. a = 0. It is easy to show (see [1]) that each Boolean algebra having 2 eleents is isoorphic with -diensional B-odul 50t. In the odul $Jl we define a further operation: Definition 2. The operation of ultiplying of a vector ae 9Jt by eleent e em is given by the rule: { o when e = 0 a when e = 1. Properties of this ultiplication are derived in [1]. Now it is possible to introduce in 9M a concept of a linear cobination: A vector ceffl is a linear cobination of the vectors aj effl»;j = 1,..., * iff there exist ej e M such that 8 c = siai + e 2 a e s a 8 = 2 e J a j i=i Vectors e(d= (1,0,...,0), e< 2 > = (0, 1, 0,..., 0),...,e< > = (0,..., 0, 1) are called base vectors of the B-odul ffl> Obviously it holds: a = («!, a 2, > «) a = E <«e(<) > where at e K. i=l 143

3 In [1] it is furtherore proved that base e(d, e(2),..., g(w) in W is unique. Definition 3. A atrix A == a\\ a\2 a 2 \ a 22 a 2n \_a \a 2... a n j where a^e M will be called Boolean atrix (or siply B-atrix) of type /n. The B-atrix 0, whose all eleents are 0 will be called the zero-b-atrix, the B-atrix, whose all eleents are 1 vill be called the unit-b-atrix, denoted by J. A atrix A is equal to a atrix B, if and only if they are of the sae type and for all i, j it holds ay = bij, where 6# e M are eleents of atrix B. The su of atrices A, B of the sae type /n is a atrix C of type /n, whose eleents satisfy c# = = ay + b^, the product of atrices A, B is a atrix D of type /n whose eleents satisfy dy = a^by, for the addition and ultiplication of eleents a^, b^ holding the sae rules as for coordinates of the B-vectores in definition 1. Definition 4. By a B-atrix decoposition of the atrix A of type \n we shall call all B-atrices A x, A 2,..., A n, for which it holds: 1. A\, A 2,..., A n are of the sae type /n; A r + A s for r + s. 2. Each Aj (j = 1,..., h) has at ost one unity in each its colun. 3. If the r-th colun of A is non-zero then the r-th colun of each Aj is non-zero. 4. A x + A A h = A. Exaple: B-atrix decoposition of A = n o i OJ [o 0 1 lj "10 10" "1" [ íoooi noio o o i oj L 1. " o o 1 1J 0011 i just 8 j the following one: " Let the B-odul ffl be given, i.e. the Boolean algebra with 2 eleents and B-odul 3Jl n, i.e. the Boolean algebra with 2 n eleents. If < n, the hooorphic apping of 9Jl w into Wd n can be considered, if ^n, also the hooorphic apping 50t onto 9W W can be considered. By the hooorphic apping of SR W onto 9W W the zero vector fro Wi is always apped onto the zero vector of B-odul 9M W, however, for the apping into this fact need not hold. For this reason we shall here consider only those hooorphiss which iage the zero vector onto zero vector. Definition 5. Let the B-atrix A of the type /n be given, whose rows are B-vectors ai,a 2,...,a ewi n. Let <pe%ft, <p = (fufi,..,/*»), fi eif. Then the atrix A represents a apping a of odul W into 9W», defined as follows: 144 a(<p) = f e2r, where y> = /i a «-

4 It is obvious that a is really the apping of Wl into9jl n, because each vector <pe9ji has only one iage in $0t n > furtherore it is evident that a(o) = o. The iage of an arbitrary vector <p e 90l w ay be deterined in the following way: behind atrix A we write vertically vector <p and add as vectors those rows atrix A, in which the verticaly written vector^has unities. This su is an iage of vector <p in odul 9M W. Exaple 1. A = B-atrix A represents apping a of odul 9W 3 into S0t 4 - Let <p = (1 1 0), then Г " thus tp = ( ) + ( ) = ( ). L! 0 For hooorphic appings the following theore is holding: Theore 1. A apping a, represented by B-atrix A of the type \n is a hooorphic apping of 9W into 9Jt w, fulfilling a(o) = o, iff the atrix A has in each colun at ost one unity. Proof: Let us denote by a\,..., a e W n rows of the atrix A. Then obviously base vector e( l ) e 9K OT is iaged onto a x, vector e( 2 ) onto az,..., d ) is iaged onto a. 1. Let the atrix a have in each colun at ost one unity. Then it holds a%a$ = o for i 4= j. Fro the definition 5 it follows a( e (0 + eu)) = a i + a j = a(e<*>) + a(e<») and furtherore, fro the properties of base vectors: Also it is iediately obvious a(e<0. ew) = a(o) = o^a i.a j = a(0. e«) = ф) = o = 0. a(e<«)\ a(l. e<«) = a(e«) = <н = 1. a(e«))j a(e<*>). a(e<'>). a(є. e«>) = є. a(e«>) consequently a is really a hooorphic apping of W into $Jl n. 2. Let a be a hooorphic apping of $Jl into 9M n, a(o) = o, and let A have at least two unities in the i-th colun, let they are in the k-th and s-th row. Then: ф(k)) = (a м, a» ak,i-i> 1, <*>k t i+l* > *,n) a(e<*>) = (a 8tli.., a 8t i_i, 1, a«.< + i,..., a 8tU ) consequently a(e(*)). a(e<*)) -4= o, because it has the i-th coordinate equal to 1, but it holds a(e( fc ). e<*)) = a(o) = o, which is a contradiction. The following theores are clear: -Al- If A is the B-atrix of the type \n and if it has in each colun and each row just one unity, then A represents the isoorphic apping of 9M onto 50l«145

5 - A2- If A is the B-atrix of the type \n, < n and if it has in each colun at ost one and in each row just one unity, then the set of iages of all vectors fro WH, i.e. oc(wl ), fors the B-odul SRn C 9K n which is isoorphic with 9H. Furtherore, we shall prove the following theore: Theore 2. If a is a hooorphic apping of 9W into 9W n > where a(o) = o, then there exists just one B-atrix A of type \n representing this apping. Proof: In [1] it is proved that the odul Wi has just one base, naely {e<i>, e< 2 >,..., e< >}. Each vector fro 9M W ay thus be expressed in the only way as a linear cobination of base vectors: (pewl => <p = I/ie<*>. Since a is the hooorphic apping, it holds: «(?>) = L7<-«(e (i) ).» Let us denote a(e^>) = at, then a% are univocally deterined rows of a B-atrix A. Theores 2 and 1 secure representation of all hooorphic appings of 9Jt into Wi n > fulfilling a(o) = o, by the set of all B-atrices of the type \n having in each colun at ost one unity. There are justi \ n ~ k B-atrices of the type \n, having h zero-coluns and n k coluns with just one unity. If we consider also the so called degenerated Boolean algebra (see [4]) having only one eleent, naely O, then we can forulate the theore: Theore 3. There exist just s = ( -f- 1 ) n hooorphic appings of a Boolean algebra with 2 eleents into a Boolean algebra with 2 n eleents, fulfilling a(6>) = 0< Corollary: There exist just r = ( + l) endoorphiss of a Boolean algebra with 2 eleents, which iage 0 onto 0. Mappings and all the ore hooorphiss, can under certain assuptions be coposed. Since this coposition is associative, there ust exist an associative operations aong B-atrices of a certain type corresponding to this coposition. Let 50t, 9W«, SOtp be B-oduls, let the B-atrix Ai of the type \n represents a apping OL\ of the B-odul 2R intp 9K n, let a atrix A 2 of the type n\p represents a apping ot 2 of 9K n into SOt^ and let us denote by a3 the coposed apping a3 = = a2ai of the odul 9K W into 3Jl p. Then the operation O corresponding to the coposition of appings is feasible only aong atrices of the types \n, n\p, the resulting atrix is of the type \p. Let the atrix A\ has eleents du, let rows of the atrix A 2 be p-diensional B-vectores ki, k 2,...,k n and let us denote the rows of the resulting atrix A$ by /i > ft >? f» which are again p-diensional B-vectors. Let the odul 9Jl has base {e<*v..., e«},. odul M n has base<{e >,.., e'*>}. Then: a 2 (e^>) =. h, oci(em) = Cii 146

6 where c* = (6n,..., Si n ). The coposed apping ot 3 = a 2 ai then iages vectors ew onto the rows of the resulting atrix A 3, thus: /, = a 3 (e«))= a 2 (ai(e(0)) = a 2 (c) = a 2 ( I J e<;>) = e5 ls a 2 (e<?>) = 2 * * 8=1 8=1 8 1 n and thus fi = L, &i s k s is the forula for the deterination of rows of the resulting B-atrix A 3. Sybolically we write A 3 --AU 0 4 Exaple 2. "0 0 r 1 0 1* x = A 2 = l 0 0_ A 3 = A г Q -4 2 = "0 1 0~ J o 1. The operation Q is a partial associative operation on the set of all B-atrices. This operations is everywhere defined on the set of all square B-atrices of the type /. The set of all square atrices of the type \ fors an algebra 21 with four operations, denoted by sybols +,., Q, T, where +,. are addition and ultiplication, defined in definition 3. With respect to this operations, 2t fors a distributive lattice. With respect to operation Q, 21 is a seigroup. The operation T is a unary operation of the transposition of J5-atrices in a way usual in the atrix-calcul. Further we consider the set of all square B-atrices of the type /, having in each row and in each colun just one unity. These atrices represent isoorphiss 2H onto 9M W, the coposition of two isoorphiss is again an isoorphis, consequently the set of these atrices, let us denote it by 2li, is closed with respect to the operation Q. 21I is a subseigroup of the seigroup 21, but 2Ii is even a group. The inverse eleent to a atrix A e 2li is the transposed atrix A 1 e 2li, unity of this group is the so called identical atrix /, representing the identical isoorphis (having unities just in the diagonal). The atrices of the algebra 2li are called B-regular. In the algebra 21 it is possible to cancellate by B-regular eleents. The atrix J is agressive with respect to operation Q for B-regular atrices. For atrices, which are not B-regular, there holds only the following inequality (in the sense of the lattice ordering): B e 21 J^J QB^B. For the right ultiplication by the eleent J even this inequality does not hold. Also the relation A Q A 1 = A 1 Q A holds only for B-regular atrices. Let us denote by 2I 2 the set of all B-atrices of the type /, having in each colun at least one unity. With respect to the addition, 2l 2 is closed. Matrices fro 2t 2 represent all seihooorphic appings of SR into 50l, fulfilling: oc(o) = 0 <x(j) = /. Again, it is clear that 2t 2 is a subseigroup of the seigroup 21 and the group 2li is a subseigroup of 2I 2. With respect to the operation T, however, 2I 2 is no ore closed. It is possible to construct further subalgebras of 21 and to investigate by eans of this etod the properties of appings of Boolean algebras. The ai of this paper was, however, only to show soe advantages of atrix representation in the investigation of hooorphiss in finite Boolean algebras. 147

7 REFERENCES [1] M t lka Jos f: Vektoriellea Modell der endlichen Booleschen Algebren (1966, Acta Univ rsitatis Palackiana Oloпшc nsis). [2] Birkhoff Garr t: Lattice Theory (A r. Math. Soci ty, N w York 1940). [3] Whit sittj. Eldon: Boolean Algebra and its Aplications. [4] Ri g r Lad.: Algebraic Methods of Matheatical Logic (Acad ia Pragu 1967). J. Chajda Přerov, třída Lidových ilicí 290 Czechoвlovakia 148