Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound

Abstract

Abstract. We show that a residually finite, congruence meet-semidistributive variety of finite type is residually < N for some finite N . This solves Pixley's problem and a special case of the restricted Quackenbush problem.

Optimal strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties

“…This condition is, for example, equivalent to congruence neutrality ( [19], [20]), or to having no covers of types 1 or 2 in congruence lattices of finite algebras in the variety (holds for locally finite varieties, [5]). It also implies the truth of Park's Conjecture, [12] (as proved by Ross Willard in [15]), and also Quackenbush's Conjecture , [13] (which holds trivially in congruence distributive case due to Jónsson's Lemma, [6], and is proved for the congruence meet-semidistributive case by Kearnes and Willard in [8]). Recently, the research in the Constraint Satisfaction Problem has shown that congruence meet-semidistributivity of the variety generated by the algebra of compatible operations is equivalent to the condition that the particular algorithm called 'localconsistency checking' would faithfully solve the Constraint Satisfaction Problem.…”

“…//the same procedure for digraphs with four two-element subalgebras // the backtracking is shortened for 8 members here // there are 15 iterations in this case // we change the limit in g_wnu3, g_3 and back_3 from 10 to 8 as explained above // for each iteration we redefine functions // set_arr8, check8 and n8_arr2 //like explained above, // that is depending on the subalgebras present set_arr8(); num3=0; num =0; num2=0; num4=0; for(i=0 /* this keeps all the permutations except for the identity one*/ perm[0] = "0132"; perm [1] = "0213"; perm [2] = "0231"; perm [3] = "0312"; perm [4] = "0321"; perm [5] = "1023"; perm [6] = "1032"; perm [7] = "1203"; perm [8] = "1230"; perm [9] = "1302"; perm [10] = "1320"; perm [11] = "2013"; perm [12] = "2031"; perm [13] = "2103"; perm [14] = "2130"; perm [15] = "2301"; perm [16] = "2310"; perm [17] = "3012"; perm [18] = "3021"; perm [19] = "3102"; perm [20] = "3120"; perm[21] = "3201"; perm[22] = "3210"; for( i=0; i<23; i++) { for(j=0; j <16; j++) aux[j] ='0'; aux [16]…”

“…// these are all the checks for the subalgebra {0,1} /* subalgebra {0,2}*/ /* ... the same as above */ for( j=0; j<4; j++) { if((arr /* value 'a' would be replaced with '0','1','2','3' as backtracking progresses */ /*we set triples of arguments for a majority term*/ arr1[0].v="012"; arr1 [1].v="013"; arr1 [2].v="021"; arr1 [3].v="023"; arr1 [4].v="031"; arr1 [5].v="032"; arr1 [6].v="102"; arr1 [7].v="103"; arr1 [8].v="120"; arr1 [9].v="123"; arr1 [10].v="130"; arr1 [11].v="132"; arr1 [12].v="201"; arr1 [13].v="203"; arr1 [14].v="210"; arr1 [15].v="213"; arr1 [16].v="230"; arr1 [17].v="231"; arr1 [18].v="301"; arr1 [19].v="302"; arr1 [20] arr2 [1].v = "002"; arr2 [2].v = "003"; arr2 [3].v = "110"; arr2 [4].v = "112"; arr2 [5].v = "113"; arr2 [6].v = "220"; arr2 [7].v = "221"; arr2 [8].v = "223"; arr2 [9].v = "330"; arr2 [10].v = "331"; arr2 [11].v = "332"; arr2 [12].v = "012"; arr2 [13].v = "013"; arr2 [14].v = "021"; arr2 [15].v = "023"; arr2 [16].v = "031"; arr2 [17].v = "032"; arr2 [18].v = "102"; arr2 [19].v = "103"; arr2 [20] int i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11, j; j=0; for(i0=0; i0<2; i0++) for(i1=0; i1<3; i1+=2) for(i2=0; i2<4; i2+=3) for(i3=0; i...…”

“….v="010"; newarray [1].v="020"; newarray [2].v="030"; newarray [3].v="101"; newarray [4].v="121"; newarray [5].v="131"; newarray [6].v="202"; newarray [7].v="212"; newarray [8].v="232"; newarray [9].v="303"; newarray [10].v="313"; newarray [11].v="323"; newarray [12].v="001"; newarray [13].v="002"; newarray [14].v="003"; newarray [15].v="110"; newarray [16].v="112"; newarray [17].v="113"; newarray [18].v="220"; newarray [19].v="221"; newarray [20]. .v="010"; newarray [1].v="020"; newarray [2].v="030"; newarray [3].v="101"; newarray [4].v="121"; newarray [5].v="131"; newarray [6].v="202"; newarray [7].v="212"; newarray [8].v="232"; newarray [9].v="303"; newarray [10].v="313"; newarray [11].v="323"; newarray [12].v="001"; newarray [13].v="002"; newarray [14].v="003"; newarray [15].v="110"; newarray [16].v="112"; newarray [17].v="113"; newarray [18].v="220"; newarray [19].v="221"; newarray [20] There are two separate source codes presented here. The first one is in sequential mode and it creates an output text file.…”

“…• we test digraphs for a wnu2-term which now means checking compatibility of a digraph given with the operations from this matrix -once a compatible operation is found we proceed to the next digraph perm [8] = "02341"; perm [9] = "02413"; perm [10] = "02431"; perm [11] = "03124"; perm [12] = "03142"; perm [13] = "03214"; perm [14] = "03241"; perm [15] = "03412"; perm [16] = "03421"; perm [17] = "04123"; perm [18] = "04132"; perm [19] = "04213"; perm [20] // the i-th permutataion of the array with the address p is now copied in the array aux, // and then we calculate the index of this copy in the array array1…”