diff --git a/src/AggregateFunctions/Combinators/AggregateFunctionCombinatorsArgMinArgMax.cpp b/src/AggregateFunctions/Combinators/AggregateFunctionCombinatorsArgMinArgMax.cpp
index 71c2bab6f6b..a1716f18725 100644
--- a/src/AggregateFunctions/Combinators/AggregateFunctionCombinatorsArgMinArgMax.cpp
+++ b/src/AggregateFunctions/Combinators/AggregateFunctionCombinatorsArgMinArgMax.cpp
@@ -68,9 +68,9 @@ public:
     String getName() const override
     {
         if constexpr (isMin)
-            return "ArgMin";
+            return nested_function->getName() + "ArgMin";
         else
-            return "ArgMax";
+            return nested_function->getName() + "ArgMax";
     }
 
     bool isState() const override { return nested_function->isState(); }
diff --git a/tests/queries/0_stateless/03127_argMin_combinator_state.reference b/tests/queries/0_stateless/03127_argMin_combinator_state.reference
new file mode 100644
index 00000000000..33482fd5fbf
--- /dev/null
+++ b/tests/queries/0_stateless/03127_argMin_combinator_state.reference
@@ -0,0 +1,12 @@
+AggregateFunction(sumArgMin, UInt64, UInt64)
+54
+0	45
+1	46
+2	47
+3	48
+4	49
+5	50
+6	51
+7	52
+8	53
+9	54
diff --git a/tests/queries/0_stateless/03127_argMin_combinator_state.sql b/tests/queries/0_stateless/03127_argMin_combinator_state.sql
new file mode 100644
index 00000000000..2eb209ed510
--- /dev/null
+++ b/tests/queries/0_stateless/03127_argMin_combinator_state.sql
@@ -0,0 +1,22 @@
+SELECT toTypeName(sumArgMinState(number, number)) FROM numbers(1);
+SELECT sumArgMinState(number, number) AS a FROM numbers(3) FORMAT Null;
+
+DROP TABLE IF EXISTS argmax_comb;
+CREATE TABLE argmax_comb(
+        id UInt64,
+        state AggregateFunction(avgArgMax, Float64, UInt64)
+    )
+    ENGINE=MergeTree() ORDER BY tuple();
+INSERT INTO argmax_comb
+    SELECT
+        CAST(number % 10, 'UInt64') AS id,
+        avgArgMaxState(CAST(number, 'Float64'), id)
+    FROM numbers(100)
+    GROUP BY id;
+SELECT avgArgMaxMerge(state) FROM argmax_comb;
+SELECT
+    id,
+    avgArgMaxMerge(state)
+FROM argmax_comb
+GROUP BY id
+ORDER BY id ASC;
\ No newline at end of file
diff --git a/tests/queries/0_stateless/03128_argMin_combinator_projection.reference b/tests/queries/0_stateless/03128_argMin_combinator_projection.reference
new file mode 100644
index 00000000000..ec9160308f2
--- /dev/null
+++ b/tests/queries/0_stateless/03128_argMin_combinator_projection.reference
@@ -0,0 +1,20 @@
+0	2024-01-01 00:00:00	2024-03-31 00:00:00
+1	2024-01-02 00:00:00	2024-04-01 00:00:00
+2	2024-01-03 00:00:00	2024-04-02 00:00:00
+3	2024-01-04 00:00:00	2024-04-03 00:00:00
+4	2024-01-05 00:00:00	2024-04-04 00:00:00
+5	2024-01-06 00:00:00	2024-04-05 00:00:00
+6	2024-01-07 00:00:00	2024-04-06 00:00:00
+7	2024-01-08 00:00:00	2024-04-07 00:00:00
+8	2024-01-09 00:00:00	2024-04-08 00:00:00
+9	2024-01-10 00:00:00	2024-04-09 00:00:00
+0	2024-01-01 00:00:00	2024-03-31 00:00:00
+1	2024-01-02 00:00:00	2024-04-01 00:00:00
+2	2024-01-03 00:00:00	2024-04-02 00:00:00
+3	2024-01-04 00:00:00	2024-04-03 00:00:00
+4	2024-01-05 00:00:00	2024-04-04 00:00:00
+5	2024-01-06 00:00:00	2024-04-05 00:00:00
+6	2024-01-07 00:00:00	2024-04-06 00:00:00
+7	2024-01-08 00:00:00	2024-04-07 00:00:00
+8	2024-01-09 00:00:00	2024-04-08 00:00:00
+9	2024-01-10 00:00:00	2024-04-09 00:00:00
diff --git a/tests/queries/0_stateless/03128_argMin_combinator_projection.sql b/tests/queries/0_stateless/03128_argMin_combinator_projection.sql
new file mode 100644
index 00000000000..f0a7f025c95
--- /dev/null
+++ b/tests/queries/0_stateless/03128_argMin_combinator_projection.sql
@@ -0,0 +1,73 @@
+DROP TABLE IF EXISTS combinator_argMin_table_r1 SYNC;
+DROP TABLE IF EXISTS combinator_argMin_table_r2 SYNC;
+
+CREATE TABLE combinator_argMin_table_r1
+(
+    `id` Int32,
+    `value` Int32,
+    `agg_time` DateTime,
+    PROJECTION first_items
+    (
+        SELECT
+            id,
+            minArgMin(agg_time, value),
+            maxArgMax(agg_time, value)
+        GROUP BY id
+    )
+)
+ENGINE = ReplicatedMergeTree('/clickhouse/tables/{database}/test_03128/combinator_argMin_table', 'r1')
+ORDER BY (id);
+
+INSERT INTO combinator_argMin_table_r1
+    SELECT
+        number % 10 as id,
+        number as value,
+        '01-01-2024 00:00:00' + INTERVAL number DAY
+    FROM
+        numbers(100);
+
+INSERT INTO combinator_argMin_table_r1
+    SELECT
+        number % 10 as id,
+        number * 10 as value,
+        '01-01-2024 00:00:00' + INTERVAL number DAY
+    FROM
+        numbers(100);
+
+SELECT
+    id,
+    minArgMin(agg_time, value),
+    maxArgMax(agg_time, value)
+FROM combinator_argMin_table_r1
+GROUP BY id
+ORDER BY id
+SETTINGS force_optimize_projection=1;
+
+-- We check replication by creating another replica
+CREATE TABLE combinator_argMin_table_r2
+(
+    `id` Int32,
+    `value` Int32,
+    `agg_time` DateTime,
+    PROJECTION first_items
+        (
+        SELECT
+            id,
+            minArgMin(agg_time, value),
+            maxArgMax(agg_time, value)
+        GROUP BY id
+        )
+)
+ENGINE = ReplicatedMergeTree('/clickhouse/tables/{database}/test_03128/combinator_argMin_table', 'r2')
+ORDER BY (id);
+
+SYSTEM SYNC REPLICA combinator_argMin_table_r2;
+
+SELECT
+    id,
+    minArgMin(agg_time, value),
+    maxArgMax(agg_time, value)
+FROM combinator_argMin_table_r2
+GROUP BY id
+ORDER BY id
+SETTINGS force_optimize_projection=1;
\ No newline at end of file