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Lattical token systems

Ovchinnikov, Sergei V.

Abstract

Stochastic token theory is a new branch of mathematical psychology. In this paper we investigate algebraic properties of token systems defined on finite lattices.

Full text

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W ) ( R$t{ y 1 $R Zt$JZ <t{ y + ? ) y bY${Y $8}j$R y ` ? ) ? qYZR ? )d - N88s 1 w >d }d1dH D yP y,K s I$R$t{ sB8R b$Y Zt$JZ {B8}j8tR Yt K 1  y  Ns${sj qB,t 7kR8R nd ]BBjjsk yP  b ;  2 O stI sjj sB8R $t \ y 7 i 7 ;  Ys% Zt$JZ {B8}j8tR Yt y I    7 ;  y 7 3 w d D N88s n yP  2 O  sjj sB8R $t \ y 7 i 7 ;  Ys% Zt$JZ {B8}j8tR stI y    7 ;  y 7  Yt  ;   _BBP 7Z}}BR  b ;   qYt Ak w d D   7 ;  y 7  y I  $8}jk$t y   y I   s {BtsI${$Bt - V js${ R $R sB8${ $P %k j8t $t R $R s `B$t BP sB8R N88s ; N R A st sB8${ js${ RZ{Y Ys %k sB8 YsR s Zt$JZ {B8}jT 8t qYt %k j8t ? ;R YsR s Zt$JZ }Rts$Bt ? )  7 ;  y 7 w 1 D PB RB8  2 O yP ? )  7 ;  y 7 stI r )  7 ; V y 7  Yt ? ` r )  7 ;   V y 7 stI ? + r )  7 ;   V y 7  _BBP 7Z}}BR ? )  7 ;  y 7 )  7 ;  1 y 7  3B stk  ;  bYs% y    7 ;  y 7 )  7 ;  1 y 7  k N88s n  ;  I qYZR  2  I  k Rk88k  )  I  ]jsjk ? ` r )  7 ;   V y 7  N F ) ? + r stI F )  7 ; . y 7  qYt  7 ; . y 7 )  7 ;  y 7 +  7 ; V y 7  3B  ; . b Ys% y    7 ; . y 7  <t{ y    7 ;  y 7 stI y    7 ; V y 7  k N88s n  ;   V  <t{ . 2   V 3B  ;   V  y    7 ;  y 7 stI y    7 ; V y 7  <t{ y    7 ; . y 7  k N88s n  ; .  <t{   V 2 .  n1 7 !%{Y$tt$,B% - qYB8 d N R A st sB8${ js${ RZ{Y Ys %k sB8 YsR s Zt$JZ {B8}j8t qYt R $R $RB8B}Y${ B Y BBjst js${ 1 Z bY Z $R Y R BP sjj sB8R $t R  _BBP k N88s n ? )  $t w 1 D RsAj$RYR st $RB8B}Y$R8 Abt R stI 1 Z  - n Ns${sj B,t RkR8R N R A s js${ 3B stk $%t y ;R b Iht PZt{$BtR * y stI  y w {P >d }nH Ak * y w ? D ) ? + y PB sjj ? ;R  y w ? D ) ? ` y PB sjj ? ;R 3 N w R , w D A s B,t RkR8 RZ{Y Ys stk + ;w $R $Y * y B  y PB RB8 y ;R  G {sjj RZ{Y s RkR8 s js${sj B,t RkR8 7$t{ Y $It$k PZt{$Bt Bt R $R tB s B,t y W )d$t * y stI y W )($t  y  VB,t + I $R s %R BP s B,t + $P PB sjj ? W ) r $t R ?+ ) r 3 r+ I ) ?3 N88s x yP * y ;w YsR s %R + ;w  Yt + )  K bY K $R st sB8 $t R stI y ) K I $R s Zt$JZ {B8}j8t BP K  _BBP 7Z}}BR + ) * K qYtd * y ) y W ) d $8}j$R y* K ) y + K )dqYZR y )ds {BtsI${$Bt<t{ + )  K PB RB8 K ;R  3B stk ?ky b Ys% ?* y ) ? + y ) y W ) ? qYZR ? ) y K ) y ` K  7$t{ d ky  ? ) d <t{ y $R s IZsj sB8 stI y ` K )d 3B stk ?cK b Ys% ? K ) ? ` K ) K W ) ? qYZR ? ) K* y ) K + y  7$t{ ( cK  ? ) ( <t{ K $R st sB8 stI y + K )( 7Z}}BR * y YsR bB %RR  K stI    G Ys% K* y ) K + y )( W ) K qYZR K )(   )   - N88s E yP  y ;w YsR s %R + ;w  Yt + ) * K bY K ) y I $R s Zt$JZ {B8}j8t BP y  _BBP qY }BBP $R IZsj B Y }%$BZR Bt WB Ys K $R s IZsj sB8 Ak N88s d - Ns ${sj qB,t 7kR8R nn qY t- YB8 PBjjBbR $88I$sjk PB8 j88sR x stI E qYB8 1 7Z}}BR stk B,t $t w R , w D YsR s %R qYt w ) \ w  y ,* y 1 D i y ; \ bY \ $R s R BP sB8R RZ{Y Ys stk y ; \ YsR s Zt$JZ {B8}j8t y I   y stI * y 1 s 8ZZsj %RR $t w R , w D  i8s, qY YB8 IBR tB {js$8 Ys RZ{Y s B,t RkR8 -$RR BB% Y s js${R b$Y tB B,t RkR8R Bt Y8 e-s8}jR $t{jZI tBt8BIZjs js${R W x stI  n  !t Y BY YstI Y $R s B,t RkR8 Bt Y tBt8BIZjs js${ W x  ] 1  qYB8 n N w R , w D A s js${sj B,t RkR8 RZ{Y Ys %k B,t YsR s %R stI PB stk ? ;R Y $R s {BtR$Rt 8RRs 8) + d NNN + E stRPB8$t ( $tB ?  qYt R $R s BBjst js${ stI w ) \ w  y ,* y 1 D i y ; \ bY Z ) \ y d , NNN ,y z i $R Y R BP sjj sB8R $t R  _BBP 7Z}}BR ? )(8)( + d NNN + E  G 8sk sRRZ8 Ys Y $R > RZ{Y Ys + 7 )  y 7 PB d  7  > yP > ) E  Yt ? ) y d `NNN` y E  !Yb$R + > Ld ) * y 1 > Le  7$t{ 8 $R {BtR$Rt y > Ld $R I$&t PB8 sjj y d , NNN ,y >  k w d D  ( + d NNN + > + > Ld ) w y d `NNN` y > D + y I > Ld ) y d `NNN` y > )( + d NNN + > 3 qYZR ? )(8 I bY 8 I $R s {BtR$Rt 8RRs BAs$tI PB8 8 Ak 8B%$t B,t + > Ld  G ,} s}}jk$t Y$R }B{RR Zt$j b BAs$t ? ) (t bY t {Bts$tR tB B,tR $t Y PB8 * y 1  qYt ? $R s `B$t BP sB8R <t{ R $R st sB8${ js${ ]jsjk w ) \ w  y ,* y 1 D i y ; \  k qYB8 d R $R s BBjst js${ - qY$R YB8 RsR Ys stk js${sj B,t RkR8 Rs$RPk$t Y bB {BtI$$BtR BP Y YB8 $R $RB8B}Y${ B Y Zt$%Rsj B,t RkR8 $tBIZ{I $t 7{$Bt d ; ]Bt{jZI$t 8s,R G Ys% RYBbt Ys ZtI RB8 R$8}j {BtI$$BtR stk js${sj B,t RkR8 $R s B,t RkR8 AsRI Bt Y BBjst js${ BP sjj RZARR BP s ht$ R qYZR s{Y B,t s{R Bt s $%t Rs C sR st sII$$Bt w RZAs{$Bt D BP s R$tj j8t B w PB8 D C  qY$R Ps{ $R $t {B8}j$st{ b$Y Y RB{YsR${ B,t YBk bY B,tR s {BtR$II sR JZstZ8 $8R BP $tPB8s$Bt stRPB8$t $tI$%$IZsj RsR iPt{R >dH p $,YB& Ns${ YBk qY$I eI$$Bt V8${st sY8s${sj 7B{$k _B%$It{ iy w dE D n; 7 !%{Y$tt$,B% >1H FT]j 3sj8st ]jsRRR BP A$tsk js$BtR sR 8I$s stI RZA8I$s GB,$t WBR =t$%R$k BP ]sj$PBt$s y%$t w dE D  >nH FT]j 3sj8st V RB{YsR${ YBk PB Y 8t{ stI Y %BjZ$Bt BP }Pt{ js$BtR sY8s${sj 7B{$sj 7{$t{R nd w dE D  EnTM; >;H FT]j 3sj8st stI FT_ *B$tBt 7B{YsR${ %BjZ$Bt BP s$Btsj$k qYT Bk stI *{$R$Bt ;n w d D  d(TdnM >xH FT]j 3sj8st 7B{YsR${ B,t YBk FBZtsj BP sY8s${sj _Rk{YBjT Bk ;d w d D  d1Td;n >EH  itb FT]j 3sj8st stI  pBP8st V RB{YsR${ 8BIj BP }Pt{ {Yst stI $R s}}j${s$Bt B d1 }R$It$sj j{$Bt }stj Iss q{Yt${sj i}B  7 xTn( ytR$Z BP sY8s${sj Ys%$Bsj 7{$t{R =t$%R$k BP ]sj$PBt$s y%$t w dx D 